
 The action of \(\operatorname{GL}(n,\mathbb {R})\operatorname{\curvearrowright }G(j,n)\)  induces transitive actions
\(\operatorname{GL}(n,\mathbb {R})\operatorname{\curvearrowright }\text{and}\quad G\operatorname{\curvearrowright }.\) 


Letting \(W_j^+:=\operatorname{span}(e_1,\ldots ,e_j)\) , \(W_j^-:=\operatorname{span}(e_n,\ldots ,e_{n-j+1})\) , the stabilizer of the flag \((W_j^\pm )_{1\le j\le n-1}\in  in \) GL(n,R)\( is \) P\(.\item The stabilizer of \) (Wj-)\( in \) G\( is \) P\(.Henceforth we identify \)  with the homogeneous space (coset space) \(G/P\)  using the orbit map \(g\mapsto g\cdot (W_j^-)\) .

The differential of the fibration \(G\rightarrow G/P\simeq  at \) eG\( yields an isomorphism\)  »/pTP(G/P)T(Wj-).

The fibration map \(G\rightarrow G/P\)  is \(P\) -equivariant w.r.t. the action of \(P\)  on \(G\)  by conjugacy and on \(G/P\)  by left translation.

The differential at \(e\in G\)  induces an isomorphism of \(P\stackrel{\operatorname{Ad}\mbox{\Large $|$\normalsize }_P}{\operatorname{\curvearrowright }}\fg /\mathfrak {p}\)  to the (isotropy) representation \(P\operatorname{\curvearrowright }T_P(G/P)\) ; here \(G\stackrel{\operatorname{Ad}}{\operatorname{\curvearrowright }} \fg \)  is the Adjoint representation and \(\operatorname{Ad}\mbox{\Large $|$\normalsize }_P\)  is the restriction to \(P\) .

 The composition \(\mathfrak {n}\hookrightarrow \fg \rightarrow \fg /\mathfrak {p}\)  is an isomorphism.

 The image \(\hat{V}_1\)  of \(V_1\subset \mathfrak {n}\)  under \(\mathfrak {n}\rightarrow \fg /\mathfrak {p}\simeq T_P(G/P)\)  is a \(P\) -invariant subspace of \(T_P(G/P)\) . This defines a \(G\) -invariant subbundle \(T(G/P)\simeq T.\item  The orbit map \) :N given by \(\alpha (g)= g\cdot (W_j^-)\) 
is an \(N\) -equivariant embedding of \(N\)  onto an open subset of \(, which we denote by \) N. The image \(\hat{N}\)  may be characterized as the collection of flags \((W_j)\)  such that \(W_j\cap W_{n-j}^+=\lbrace 0\rbrace \)  for every \(1\le j\le n-1\) .

The embedding \(\alpha \)  is contact with respect to the subbundles \(V_1\subset TN\)  and \(T(G/P)\) .

 For every \(r\in (0,\infty )\)  we have \(\alpha \circ \delta _r=\hat{\delta }_r\circ \alpha \) , where \(\hat{\delta }_r: is given by \) r(x)=grx\( with \) gr=diag(r-1,...,r-n)\(.\)
(REF )-(REF ) follow readily from linear algebra or basic theory of manifolds.
(REF ). To see this, it suffices to show that the image of \(V_1\subset \mathfrak {n}\)  under \(\mathfrak {n}\hookrightarrow \operatorname{\fg \mathfrak {l}}(n,\mathbb {R})/\mathfrak {p}^-\)  is invariant under \(\operatorname{Ad}\mbox{\Large $|$\normalsize }_{P^-}\) . Since \(P^-\)  is connected, it therefore suffices to see that \([\mathfrak {p}^-,V_1]\subset V_1+\mathfrak {p}^-\) ; this follows readily by applying (REF ).
(). The orbit map \(\alpha :N\rightarrow  is smooth and \) N\(-equivariant. Since the differential of \)\( at \) eN\( is an isomorphism \) D(e):n »/p\(, the map \)\( is an \) N\(-equivariant immersion onto an open subset. But \) NP-={e}\( so it is also injective, and hence \)\( is an embedding.\) Let \(Z:=\lbrace (W_j)\in W_j\cap W_{n-j}^+=\lbrace 0\rbrace \;\text{for all}\;1\le j\le n-1\rbrace \) . If \((W_j)\)  belongs to the orbit \(\hat{N}=N\cdot (W_j^-)\)  then \((W_j)\in Z\)  since \(W_j^-\cap W_{n-j}^+=\lbrace 0\rbrace \)  and \(N\)  fixes \(W_{n-j}^+\)  for all \(j\) . Now suppose \((W_j)\in Z\) . We prove by induction on \(k\)  that the \(N\) -orbit of \((W_j)\)  contains a flag \((W_j^k)\)  such that \(W_j^k=W_j^-\)  for all \(j\le k\) . Since \(W_1\cap W_{n-1}^+=\lbrace 0\rbrace \)  we may choose \(v=(v_1,\ldots ,v_n)\in W_1\)  with \(v_n=1\) . If \(n\in N\)  is the block matrix
\(\left[\begin{matrix}I& -b\\0&1\end{matrix}\right]\) 
with \(b=[v_1,\ldots ,v_{n-1}]^t\)  we have \(n\cdot v\in W_1^-\) , so letting \((W_j^1):=n\cdot (W_j)\)  we have \(W_1^1=W_1^-\) , establishing the \(k=1\)  case. The inductive step follows similarly, by working in the quotient \(\mathbb {R}^n/W_{k-1}^-\) .
(11). Both \(V_1\)  and \( are \) N\(-invariant subbundles, and \)\( is \) N\(-equivariant, so the contact property follows from the fact that \) P)TP(G/P)\( is the image of \) V1n\( under \) D(e)\(.\) (REF ). Since \(g_r\in P^-\)  we have
\(\alpha (\delta _r(g))=\delta _r(g)\cdot P=g_rgg_r^{-1}\cdot P=g_rg\cdot P=\hat{\delta }_r(\alpha (g))\,.\) 

Automorphisms
We collect some properties of automorphisms of \(G\)  and graded automorphisms of \(N\) .
Lemma 2.3 (Properties of \(\operatorname{Aut}(G)\) )


Transpose inverse \(T\mapsto (T^t)^{-1}\)  descends to a Lie group automorphism of \(G\) . The group \(\operatorname{Aut}(G)\)  of Lie group automorphisms of \(G\)  is generated by inner automorphisms and transpose inverse.

Let \(\tau :\operatorname{GL}(n,\mathbb {R})\rightarrow \operatorname{GL}(n,\mathbb {R})\)  be the automorphism given by
\(\tau (T)=\Pi (T^t)^{-1}\Pi ^{-1}\) 
where \(\Pi \in \operatorname{GL}(n,\mathbb {R})\)  is the permutation matrix with \(\Pi e_i=e_{n-i+1}\) . Then the induced automorphism \(\operatorname{\fg \mathfrak {l}}(n,\mathbb {R})\rightarrow \operatorname{\fg \mathfrak {l}}(n,\mathbb {R})\)  – which we also denote by \(\tau \)  by abuse of notation – is given by
\((\tau (A))_{i,j}=-A_{n-j+1,n-i+1}\,.\) 
Then \(\tau \)  preserves \(N\) , and induces a graded automorphism \(\mathfrak {n}\rightarrow \mathfrak {n}\) .

The automorphism \(\tau :\operatorname{GL}(n,\mathbb {R})\rightarrow \operatorname{GL}(n,\mathbb {R})\)  descends to an automorphism \(G\rightarrow G\) , which we also denote by \(\tau \)  (by further abuse of notation).


 
(1). This follows from Theorem REF  since the only automorphism of \(\mathbb {R}\)  is the identity map.
Therefore there exists \(\Phi _1\in \operatorname{Aut}(G)\)  in the group generated by inner automorphisms and transpose-inverse, such that \(D\Phi _1=D\Phi \) . Hence \(\Phi _2:=\Phi _1^{-1}\circ \Phi \in \operatorname{Aut}(G)\)  is the identity on the connected component of \(e\in G\) . By considering centralizers of elements \(g\in G\)  of order 2, it is not hard to see that \(\Phi _2=\operatorname{id}\) , and hence \(\Phi =\Phi _1\) .
(2) and (3) are straightforward.
We now consider the group \(\operatorname{Aut}_{gr}(N)\)  of graded automorphisms of \(N\) .
Lemma 2.5 (Properties of \(\operatorname{Aut}_{gr}(N)\) ) 

Assume \(n\ge 4\) .

 If \(\Phi \in \operatorname{Aut}(G)\)  and \(\Phi (N)=N\) , then \(\Phi \)  induces a graded automorphism of \(N\)  if and only if \(\Phi \mbox{\Large $|$\normalsize }_N=\tau ^\varepsilon \circ I_g\mbox{\Large $|$\normalsize }_N\)  for some \(\varepsilon \in \lbrace 0,1\rbrace \) , \(g=\operatorname{diag}(\lambda _1,\ldots ,\lambda _n)\in G\) .

Every graded automorphism of \(N\)  arises as in (1).

For \(1\le j\le n-1\)  let \(\mathfrak {k}_j\subset \mathfrak {n}\)  be the Lie subalgebra generated by \(\lbrace X_{i,i+1}\rbrace _{i\ne n-j}\) , and \(K_j\subset N\)  be the Lie subgroup with Lie algebra \(\mathfrak {k}_j\) .
A graded automorphism \(N\rightarrow N\)  is induced by conjugation by some \(g=\operatorname{diag}(\lambda _1,\ldots ,\lambda _n)\)  if and only if it preserves the subgroups \(K_j\)  for \(1\le j\le n-1\) .


(REF ). Suppose \(\Phi =\tau ^\varepsilon \circ I_g\)  for some \(g=\operatorname{diag}(\lambda _1,\ldots ,\lambda _n)\) . Note that \(I_g\)  induces a graded automorphism of \(N\)  since
\(I_g\circ \delta _r=I_g\circ I_{g_r}=I_{gg_r}=I_{g_rg}=I_{g_r}\circ I_g=\delta _r\circ I_g\,.\) 
Combining with Lemma REF (2) we get that \(\Phi \)  induces a graded automorphism of \(N\) , proving the “if” direction.
Now suppose \(\Phi \in \operatorname{Aut}(G)\)  preserves \(N\) , and induces a graded automorphism \(N\rightarrow N\) .
By Lemma REF (1)
we have \(\Phi =\tau ^{\varepsilon }\circ I_g\)  for some \(\varepsilon \in \lbrace 0,1\rbrace \) , \(g\in G\) . By Lemma REF (2), after postcomposing with \(\tau \)  if necessary, we may assume without loss of generality that \(\Phi =I_g\)  is an inner automorphism. The condition \(\Phi (N)=N\)  is equivalent to saying that \(g\)  belongs to the normalizer of \(N\)  in \(G\) . This is just the image of \(P^+\)  under the quotient map \(\operatorname{GL}(n,\mathbb {R})\rightarrow G\) . (To see this, note that if \(\hat{g}\in \operatorname{GL}(n,\mathbb {R})\)  is a lift of \(g\) , then \(\hat{g}\)  normalizes \(N\subset \operatorname{GL}(n,\mathbb {R})\) . But \(\operatorname{span}(e_1)\)  is the unique fixed point of the action \(N\operatorname{\curvearrowright }G(1,n)\) , so \(\hat{g}\)  fixes \(\operatorname{span}(e_1)\) ; passing to the quotient \(\mathbb {R}^n/\operatorname{span}(e_1)\)  and arguing by induction, one gets that \(\hat{g}\in P^+\) . )
Furthermore, after multiplying \(g\)  by a diagonal matrix, we may assume without loss of generality that \(g\in N\) .
Now \(I_g:N\rightarrow N\)  is a graded automorphism, so for all \(r>0\)  we have \(I_g\circ \delta _r=\delta _r\circ I_g\)  , i.e.
\(&I_g\circ I_{g_r}=I_{g_r}\circ I_g\\\Rightarrow &I_{g_rgg_r^{-1}g^{-1}}=\operatorname{id}_N\\\Rightarrow &g_rgg_r^{-1}g^{-1}\in \operatorname{Center}(N)\\\Rightarrow &g_rXg_r^{-1}-X\in \operatorname{Center}(\mathfrak {n})\) 

where \(g=\exp X\)  for \(X\in \mathfrak {n}\) ; here we are using the fact that the exponential \(\exp :\mathfrak {n}\rightarrow N\)  is a diffeomorphism. Calculating in \(\operatorname{\fg \mathfrak {l}}(n,\mathbb {R})\)  and letting \(X=\sum _{j>i}a_{ij}X_{ij}\)  we have
\(g_rXg_r^{-1}-X=\sum _{j>i}a_{ij}(r^{j-i}-1)X_{ij}\in \operatorname{Center}(\mathfrak {n})\,.\) 
Therefore \(X\in \operatorname{Center}(\mathfrak {n})\) , and \(g=\exp X\in \operatorname{Center}(N)\) , and \(I_g=\operatorname{id}\) .
(2). Let \(\phi :\mathfrak {n}\rightarrow \mathfrak {n}\)  be a graded automorphism of \(\mathfrak {n}\) . We will use the fact that for any \(x\in V_1\)  and any \(j\ge 1\) , we have \(\phi \circ (\operatorname{ad}x\mbox{\Large $|$\normalsize }_{V_j})=(\operatorname{ad}\, \phi x\mbox{\Large $|$\normalsize }_{V_j})\circ \phi \) , and in particular
\(\operatorname{rank}(\operatorname{ad}\, x|_{V_j})=\operatorname{rank}(\operatorname{ad}\, \phi x|_{V_j})\) .
First suppose \(n=4\) .
Set
\(\begin{aligned}X_0=X _{23}\,,\;X_1=X _{12}\,,\;X_2=X _{34}\,,\\Y_1=X _{13}\,,\;Y_2=-X _{24}\,,\;Z=X _{14}\end{aligned}\) 
Then the following are the only nontrivial bracket relations between the basis elements:
\(\begin{aligned}~[X_0,X_1]=-Y_1, \;\; [X_0, X_2]=-Y_2,\\[X_1, Y_2]=-Z=[X_2, Y_1].\end{aligned}\) 
Let \(W_0:=\operatorname{span}(X_0)\) , and \(W:=\operatorname{span}(X_1, X_2)\) . If \(x\in V_1\)  then \(\operatorname{rank}(\operatorname{ad}x\mbox{\Large $|$\normalsize }_{V_1})\le 1\)  iff \(x\in W\) , and \(\operatorname{rank}(\operatorname{ad}x\mbox{\Large $|$\normalsize }_{V_2})=0\)  iff \(x\in W_0\) . It follows that \(\phi (W)=W\)  and \(\phi (W_0)=W_0\) , and so
\(\phi X_0=a_0X_0\,,\quad \phi X_1=a_{11}X_1+a_{21}X_2\,,\quad \phi X_2=a_{12}X_1+a_{22}X_2\) 
for some \(0\ne a_0\in \mathbb {R}\) , \((a_{ij})\in \operatorname{GL}(2,\mathbb {R})\) . Hence
\(\phi Y_1&=[\phi X_1, \phi X_0]=a_0a_{11}Y_1+a_0a_{21}Y_2\\\phi Y_2&=[\phi X_2, \phi X_0]=a_0a_{12}Y_1+a_0a_{22}Y_2\\ 0&=[\phi X_1,\phi Y_1]=-2a_0a_{11}a_{21}Z\\0&=[\phi X_2,\phi Y_2]=-2a_0a_{12}a_{22}Z.\) 

Case 1. \(a_{11}\ne 0\) . By (REF ) we have \(a_{21}=0\) , and since \(\det (a_{ij})\ne 0\)  it follows that \(a_{22}\ne 0\) , and so \(a_{12}=0\)  by (). Thus \(\phi \mbox{\Large $|$\normalsize }_{V_1}\)  is diagonal in the basis \(X_0,X_1,X_2\) . Hence it is induced by an inner automorphism \(I_g:G\rightarrow G\) , where \(g\)  is diagonal.
Case 2. \(a_{11}=0\) . Since \(\det (a_{ij})\ne 0\)  it follows that \(a_{12}\)  and \(a_{21}\)  are both nonzero. Then () gives \(a_{22}=0\) . Therefore modulo post-composition with the automorphism \(\tau \)  we get that \(\phi \mbox{\Large $|$\normalsize }_{V_1}\)  is diagonal, and so (2) holds.
Now suppose \(n\ge 5\) .
First observe that
\(\lbrace x\in V_1\mid \text{rank}(\text{ad}\; x|_{V_1})=1\rbrace =(\mathbb {R}X_{1,2}\cup \mathbb {R}X_{n-1,n})\backslash \lbrace 0\rbrace .\) 
So \(\phi \)  either preserves or switches the two lines \(\mathbb {R}X_{1,2}\) , \(\mathbb {R}X_{n-1,n}\) ,
and therefore after postcomposing with \(\tau \)  if necessary, we may assume without loss of generality that \(\phi \)  preserves them.
Then we observe that
\(\lbrace x\in V_1\mid \text{rank}(\text{ad}\; x|_{V_{n-2}})=0\rbrace =\text{span}\lbrace X_{2,3}, \cdots , X_{n-2, n-1}\rbrace .\) 
So \(\phi (\mathfrak {h}_0)=\mathfrak {h}_0\) , where \(\mathfrak {h}_0\)  is the Lie subalgebra
of \(\mathfrak {n}\)  generated by
\(X_{2,3}, \cdots , X_{n-2, n-1}\) . Note \(\mathfrak {h}_0\)  is isomorphic to \(\mathfrak {n}_{n-2}\) , i.e. the strictly upper triangular matrices in \(\operatorname{\fg \mathfrak {l}}(n-2,\mathbb {R})\) .
Suppose \(n=5\) . In this case \(\mathfrak {h}_0{\cap V_1}=\text{span}\lbrace X_{2,3}, X_{3,4}\rbrace \) . We have
\(\lbrace x\in \mathfrak {h}_0\cap V_1\mid [x, X_{1,2}]=0\rbrace =\mathbb {R}X_{3,4}\) 
and
\(\lbrace x\in \mathfrak {h}_0\cap V_1\mid [x, X_{4,5}]=0\rbrace =\mathbb {R}X_{2,3}.\) 
Because \(\phi \)  preserves the two lines \(\mathbb {R}X_{1,2}\) , \(\mathbb {R}X_{4,5}\)  then it also preserves the two lines \(\mathbb {R}X_{2,3}\) , \(\mathbb {R}X_{3,4}\) . Hence \(\phi \)  is diagonal in the basis \(\lbrace X_{i,i+1}\rbrace _{1\le i\le 4}\) , and therefore (2) holds.
Now we assume \(n\ge 6\)  and that (2) holds for \(\mathfrak {n}_k\)  with \(k\le n-2\) .
We already observed \(\phi (\mathfrak {h}_0)=\mathfrak {h}_0\) , and have reduced to the case that the two lines \(\mathbb {R}X_{1,2}\) , \(\mathbb {R}X_{n-1,n}\)  are preserved by \(\phi \) . Since \(\mathfrak {h}_0\)  is isomorphic to \(\mathfrak {n}_{n-2}\) , the induction hypothesis implies that either
\(\phi (\mathbb {R}X_{i, i+1})=\mathbb {R}X_{i, i+1}\)  for all \(i=2,\cdots , n-2\)  or
\(\phi (\mathbb {R}X_{i, i+1})=\mathbb {R}X_{n-i, n+1-i}\)  for all \(i=2,\cdots , n-2\) .
Since \([X_{1,2}, X_{2,3}]\ne 0\)  and
\([X_{1,2}, X_{n-2,n-1}]=0\) , and \(\phi \)  is an automorphism of Lie algebra, we see that
\(\phi (\mathbb {R}X_{i, i+1})=\mathbb {R}X_{i, i+1}\)  for all \(i=1,\cdots , n-1\) ,
and so (2) holds.
(3). Let \(\Phi \)  be a graded automorphism, so by (2) we have \(\Phi =\tau ^{\varepsilon }\circ I_g\)  for \(\varepsilon \in \lbrace 0,1\rbrace \)  and some \(g\)  diagonal. If \(\varepsilon =0\)  then \(D\Phi (X_{i,i+1})\in \mathbb {R}X_{i,i+1}\) , so \(\Phi \)  preserves \(K_j\)  for all \(j\) . On the other hand, by (REF ) we have \(\tau (K_j)=K_{n-j}\) , so if \(\Phi (K_j)=K_j\)  for all \(1\le j\le n-1\)  then \(\varepsilon =0\) .
We now consider the action of \(\operatorname{Aut}(G)\)  on the flag manifold.
Lemma 2.7


There is a 1-1 correspondence between cosets of \(P\)  and their stabilizers in \(G\)  with respect to the action \(G\operatorname{\curvearrowright }G/P\) : if \(g_1,g_2\in G\)  then
\(\operatorname{Stab}(g_1P)=\operatorname{Stab}&(g_2P)\quad \iff \quad g_1Pg_1^{-1}=g_2Pg_2^{-1}\\&\iff g_1P=g_2P\,.\) 


Every \(\Phi \in \operatorname{Aut}(G)\)  permutes the conjugates of \(P\) ; by (1) we thereby obtain an action of \(\operatorname{Aut}(G)\stackrel{\rho }{\operatorname{\curvearrowright }}G/P\) 
defined by
\(\operatorname{Stab}(\Phi \cdot gP)=\Phi (\operatorname{Stab}(gP))\,.\) 
More explicitly, if \(\Phi (P)=hPh^{-1}\)  then
\(\Phi \cdot gP=\Phi (g)hP\,;\) 
in particular \(\rho (\Phi )\)  defines a smooth diffeomorphism.

For every \(\Phi \in \operatorname{Aut}(G)\)  the map
\(\rho (\Phi ):G/P\longrightarrow G/P\) 
is \(G\) -equivariant from \(G\stackrel{\ell }{\operatorname{\curvearrowright }}G/P\)  to \(G\stackrel{\ell _\Phi }{\operatorname{\curvearrowright }}G/P\) , where \(\ell (\bar{g})( gP):=\bar{g}gP\)  and \(\ell _\Phi (\bar{g})(gP):=\Phi (\bar{g})gP\) .

The action \(\operatorname{Aut}(G)\stackrel{\rho }{\operatorname{\curvearrowright }} G/P\)  preserves the horizontal subbundle \(T(G/P)\) .

If \(\Phi _0\in \operatorname{Aut}(G)\)  is transpose-inverse, then \(\Phi _0\cdot (W_j)=(W_j^\perp )\)  for every \((W_j)\in , i.e. \) (0)=\(.\)
(1). Note that the normalizer of \(P\)  in \(G\)  is \(P\)  itself. Therefore
\(\operatorname{Stab}&(g_1P)=\operatorname{Stab}(g_2P)\quad \iff \quad g_1Pg_1^{-1}=g_2Pg_2^{-1}\\&\iff (g_2^{-1}g_1)P(g_2^{-1}g_1)^{-1}=P\quad \iff \quad g_2^{-1}g_1\in P\\&\iff g_1P=g_2P\,.\) 

(2). Note that \(\tau (P)=P\)  so \(\tau \)  permutes the conjugates of \(P\) ; by Lemma REF (1) it follows that every \(\Phi \in \operatorname{Aut}(G)\)  permutes the conjugates of \(P\) . Since \(\operatorname{Stab}(gP)=gPg^{-1}\) , if \(\Phi (P)=hPh^{-1}\) , then for every \(g\in G\)
\(\operatorname{Stab}(\Phi \cdot gp)=\Phi (\operatorname{Stab}(gP))=\Phi (gPg^{-1})\\=\Phi (g)hPh^{-1}(\Phi (g))^{-1}=\operatorname{Stab}(\Phi (g)hP)\) 

so \(\Phi \cdot gP=\Phi (g)hP\) .
(3). For \(g,\bar{g}\in G\) , by (2) we have
\(\Phi \cdot (\ell (\bar{g})\cdot gP)=\Phi \cdot (\bar{g}gP)=\Phi (\bar{g}g)hP\\=\ell (\Phi (\bar{g}))\cdot (\Phi (g)hP)=\ell _\Phi (\bar{g})\cdot (\Phi \cdot gP)\) 

as claimed.
(4). If \(\Phi =I_{\bar{g}}\)  is an inner automorphism, then \(I_{\bar{g}}\cdot gP=\bar{g}gP\)  by (REF ); hence \(I_{\bar{g}}\cdot \)  preserves \( by Lemma~\ref {lem_properties_of_f}(9). In the case \) =\(, we have \) (P)=P\(, so by (\ref {eqn_aut_g_action_def_2}) we have \) gP=(g)P\(, and since \) (Xi,i+1)=Xn-i,n-i+1\( we have \) (V1)=V1\(. It follows that the differential of \) : G/PG/P\( at \) P\( preserves \) P)\(. Using (3) and the fact that the actions \)\( and \)\( preserve \) , we conclude that \(\tau \cdot \)  preserves \(. By Lemma~\ref {lem_properties_aut_g} the group \) Aut(G)\( is generated by \)\( and the inner automorphisms, so (4) follows.\)
(5). Let \(\Pi \in \operatorname{GL}(n,\mathbb {R})\)  be the permutation matrix with \(\Pi e_i=e_{n-i}\) , so \(\Phi _0(P)=\Pi P\Pi ^{-1}\) . If \((W_j)=g\cdot (W_j^-)\) , then using (REF ) we have
\(\Phi _0\cdot (W_j)&=\Phi _0\cdot (g\cdot (W_j^-))=(\Phi _0(g)\Pi )\cdot (W_j^-)\\&=((g^{-1})^t\Pi W_j^-)=((g^{-1})^tW_j^+)\,.\) 

Since \((g^{-1})^tW_j^+\)  is orthogonal to \(gW_{n-j}^-\)  for every \(1\le j\le n-1\) , assertion (5) follows.

Fibrations between flag manifolds
We now consider partial flags and the associated flag manifolds.
If \(\Sigma \subset \lbrace 1,\ldots ,n-1\rbrace \) , we let \({\Sigma }\)  be the collection of partial flags \((W_j)_{j\in \Sigma }\)  where \(W_j\subset \mathbb {R}^n\)  has dimension \(j\) . Then, as in the case when \(\Sigma =\lbrace 1,\ldots ,n-1\rbrace \) , the set \({\Sigma }\)  is a smooth submanifold of the product \(\prod _{j\in \Sigma }G(j,n)\)  of Grassmannians, and the actions \(G\operatorname{\curvearrowright }G(j,n)\)  yield a transitive smooth action \(G\operatorname{\curvearrowright }{\Sigma }\) . Taking \((W_j^-)_{j\in \Sigma }\)  to be the basepoint, and letting \(P_{\Sigma }=\operatorname{Stab}((W_j^-)_{j\in \Sigma })\subset G\)  be its stabilizer, we obtain a \(G\) -equivariant diffeomorphism \(G/P_{\Sigma }\rightarrow {\Sigma }\) ; we identify \({\Sigma }\)  with \(G/P_{\Sigma }\)  accordingly. We will be interested in the case when \(\Sigma =\lbrace j\rbrace \)  for some \(j\in \lbrace 1,\ldots ,n-1\rbrace \) , so to simpify notation we let \(j:={\lbrace j\rbrace }=G(j,n)\) .
Lemma 2.10


If \(\Sigma _1\subset \Sigma _2\subset \lbrace 1,\ldots ,n-1\rbrace \)  then we obtain a smooth \(G\) -equivariant fibration \(\pi _{\Sigma _1,\Sigma _2}:{\Sigma _2}\rightarrow {\Sigma _1}\)  by “forgetting subspaces”, i.e. \(\pi _{\Sigma _1,\Sigma _2}((W_j)_{j\in \Sigma _2}):=(W_j)_{j\in \Sigma _1}\) .

The fiber of \(\pi _{\Sigma _1,\Sigma _2}\)  passing through \((W_j^-)\)  is the \(P_{\Sigma _1}\)  orbit \(P_{\Sigma _1}\cdot (W_j-)\) , which is diffeomorphic to \(P_{\Sigma _1}/P_{\Sigma _2}\) .
To simplify notation, for \(j\in \lbrace 1,\ldots ,n-1\rbrace \)  we let \(j:={\lbrace j\rbrace }=G(j,n)\)  and
\(\pi _j:=\pi _{\lbrace j\rbrace ,\lbrace 1,\ldots ,n-1\rbrace }:{\lbrace 1,\ldots ,n-1\rbrace }\rightarrow j\,,\) 
so the fiber of \(\pi _j\)  passing through the basepoint \(P\)  is \(P_jP\) .

The intersection of the fiber \(\pi _j^{-1}(W_j^-)\)  with \(\hat{N}=N\cdot (W_j^-)\)  is \(K_j\cdot (W_j^-)\) , where \(K_j\subset N\)  is as in Lemma REF (3). Hence the orbit map \(\alpha :N\rightarrow \hat{N}\)  carries the coset foliation of \(K_j\)  to the foliation defined by \(\pi _j\) .


(1) and (2) are a special case of a general fact: if \(K\)  is a Lie group and \(H_1\subset H_2\subset K\)  are closed subgroups, the quotient map \(K/H_1\rightarrow K/H_2\)  is a \(K\) -equivariant fibration; fiber passing through \(H_1\)  is \(H_2H_1\subset K/H_1\)  and it is diffeomorphic to \(H_2/H_1\) .
(4). The subbundle of \(T(G/P)\)  tangent to the fibers of the fibration \(\pi _j:G/P\rightarrow G/P_j\)  is \(G\) -invariant, and its value at the basepoint \(P\in G/P\)  is the subspace \(\mathfrak {p}_j/\mathfrak {p}\) . The subbundle of \(TN\)  tangent to the coset foliation of \(K_j\)  maps under the orbit map \(\alpha :N\rightarrow \hat{N}\)  to an \(N\) -invariant subbundle of \(T\hat{N}\)  whose value at \(P\)  is the subspace \((\mathfrak {k}_j+\mathfrak {p})/\mathfrak {p}\) . But \(\mathfrak {k}_j+\mathfrak {p}=\mathfrak {p}_j\) , so the two subbundles are the same. It follows that \(\pi _j^{-1}(W_j^-)\cap \hat{N}\)  is the union of a closed set of \(K_j\) -orbits in \(\hat{N}\) ; however, \(\pi _j^{-1}(W_j)\cap \hat{N}\)  is invariant under the action of \(\hat{\delta }_r\) , so it can contain only one \(K_j\) -orbit.


Dynamics of Carnot dilations
 
The following result is a special case of [1]}; we give a self-contained proof for the reader's convenience.
Lemma 2.11 
Pick \(1\le j\le n-1\) .

Let \(K_1\subset 1\)  be a compact subset disjoint from \(Z_1:=\lbrace W_1\in 1\mid W_1\subset W_{n-j}^+\rbrace \) ,
and \(U_1\subset 1\)  be an open subset containing \(Z_2:=\lbrace W_1\in 1\mid W_1\subset W_j^-\rbrace \) . Then there exists \(\bar{r}_1=\bar{r}_1(K_1,U_1)>0\)  such that for every \(r\le \bar{r}_1\)  we have \(\hat{\delta }_r(K_1)\subset U_1\) .

Let \(K\subset j\)  be a compact subset such that \(W_j\cap W_{n-j}^+=\lbrace 0\rbrace \)  for every \(W_j\in K\) , and \(U\subset j\)  be an open subset containing \(W_j^-\) . Then there exists \(\bar{r}=\bar{r}(K,U)>0\)  such that for every \(r\le \bar{r}\)  we have \(\hat{\delta }_r(K)\subset U\) .


(1). Let \(\pi _{W_{n-j}^+}\)  and \(\pi _{W_j^-}\)  be the projections onto the summands of the decomposition \(\mathbb {R}^n=W_{n-j}^+\oplus W_j^-\) .
Define \(\hat{\beta }:\mathbb {R}^n\setminus W_{n-j}^+\rightarrow [0,\infty )\)  by \(\hat{\beta }(v):=\frac{\Vert \pi _{W_{n-j}^+}v\Vert }{\Vert \pi _{W_j^-}v\Vert }\) . Note that by homogeneity \(\hat{\beta }\)  descends to \(\beta :1\setminus Z_1\rightarrow [0,\infty )\) .
Since \(K_1\)  is compact and \(U_1\)  is open, we may choose \(C_2\le C_1<\infty \)  such that \(\beta (K_1)\subset [0,C_1]\)  and \(\beta ^{-1}([0,C_2])\subset U\) .
If \(v\in \mathbb {R}^n\setminus W_{n-j}^+\)  and \(r\le 1\)  then
\(\Vert \pi _{W_{n-j}^+}(\hat{\delta }_rv)\Vert &=\Vert \sum _{i=1}^{n-j}r^{-i}v_i\Vert \le r^{-(n-j)}\Vert \pi _{W_{n-j}^+}v\Vert \\\Vert \pi _{W_j^-}(\hat{\delta }_rv)\Vert &=\Vert \sum _{i=n-j+1}^nr^{-i}v_i\Vert \ge r^{-(n-j+1)}\Vert \pi _{W_j^-}v\Vert \\\Rightarrow &\hat{\beta }(\hat{\delta }_rv)\le r\hat{\beta }(v)\,.\) 

Hence for \(W_1\in 1\setminus Z_1\)  and \(r\le 1\)  we have \(\beta (\hat{\delta }_rW_1)\le r\beta (W_1)\) .
Taking \(\bar{r}_1:=C_2/C_1\) , if \(r\le \bar{r}_1\)  then
\(\hat{\delta }_r(K_1)\subset \hat{\delta }_r(\beta ^{-1}([0,C_1]))\subset \beta ^{-1}([0,C_2])\subset U_1\,.\) 
(2). Note that
\(K_1:=\lbrace W_1\in 1\mid \exists W_j\in K\;\text{s.t.}\;W_1\subset W_j\rbrace \) 
is a compact subset of \(1\setminus Z_1\) , and there is an open subset \(U_1\subset 1\)  containing \(Z_2\)  such that
if \(W_j\in j\)  and \(\lbrace W_1\in 1\mid W_1\subset W_j\rbrace \subset U_1\) , then \(W_j\in U\) .
Let \(\bar{r}=\bar{r}_1(K_1,U_1)\)  be as in (1), and choose \(W_j\in K\) , \(r\le \bar{r}\) . Then by the choice of \(r\) , for every \(W_1^{\prime }\in 1\)  with \(W_1^{\prime }\subset W_j\)  we have \(\hat{\delta }_r(W_1^{\prime })\in U_1\) . Therefore \(\lbrace W_1\in 1\mid W_1\subset \hat{\delta }_r(W_j)\rbrace \subset U_1\) , and hence \(\hat{\delta }_r(W_j)\in U\) , by our assumption on \(U_1\) .


Sobolev mappings and the Pullback Theorem
 
We give a very brief discussion here, and refer the reader to [2]} and the references therein for more details.
Let \(H\) , \(H^{\prime }\)  be Carnot groups with gradings \(\mathfrak {h}=\oplus _jV_j\) , \(\mathfrak {h}^{\prime }=\oplus _j V_j^{\prime }\) , Carnot dilations denoted by \(\delta _r\) , and Carnot-Caratheodory distance functions \(d_{CC}\) , \(d_{CC}^{\prime }\) . Let \(\nu =\sum _jj\dim V_j\)  be the homogeneous dimension of \(H\) .
Choose \(p>\nu \) . If \(U\subset H\)  is open and \(f:U\rightarrow H^{\prime }\)  is a continuous map then \(f\)  is a \(W^{1,p}_{\operatorname{loc}}\) -mapping if there exists \(g\in L^p_{\operatorname{loc}}(U)\)  such that for every 1-Lipschitz function \(\psi :H^{\prime }\rightarrow \mathbb {R}\)  and every unit length \(X\in V_1\) , the distribution derivative \(X(\psi \circ f)\)  satisfies \(|X(\psi \circ f)|\le g\)  almost everywhere. If \(f:H\supset U\rightarrow U^{\prime }\subset H^{\prime }\)  is a quasiconformal homeomorphism then \(f\)  is a \(W^{1,p}_{\operatorname{loc}}\) -mapping for some \(p>\nu \) .
If \(f:U\rightarrow H^{\prime }\)  is a \(W^{1,p}_{\operatorname{loc}}\) -mapping then for a.e. \(x\in U\)  the map \(f\)  has a Pansu differential, i.e. letting \(f_x:=\ell _{f(x)^{-1}}\circ f\circ \ell _x\) , \(f_{x,r}:= \delta _r^{-1}\circ f_x\circ \delta _r\)  there is a graded group homomorphism \(D_Pf(x):H\rightarrow H^{\prime }\)  such that
\(f_{x,r}\stackrel{C^0_{\operatorname{loc}}}{\longrightarrow } D_Pf(x)\) 
as \(r\rightarrow 0\) . A Lie group homomorphism \(\Phi : H\rightarrow H^{\prime }\)  is graded if \(\delta _r\circ \Phi =\Phi \circ \delta _r\)  for all \(r>0\) . We will often use \(D_Pf(x)\)  to denote the induced graded homomorphism of Lie algebras \(\mathfrak {h}\rightarrow \mathfrak {h}^{\prime }\) .
We let \(\lbrace X_j\rbrace _{1\le j\le \dim H}\)  be a graded basis for \(\mathfrak {h}\) , and \(\lbrace \theta _j\rbrace _{1\le j\le \dim H}\)  be the dual basis.
The weight of a subset \(I\subset \lbrace 1,\ldots ,\dim H\rbrace \)  is given by
\(\operatorname{wt}I:=-\sum _{i\in I}\deg i\) 
where \(\deg :\lbrace 1,\ldots ,\dim H\rbrace \rightarrow \lbrace 1,\ldots , s\rbrace \)  is defined by \(\deg i=j\)  iff \(X_i\in V_j\) . For a non-zero left-invariant form
\(\alpha = \sum _{I} a_I \theta _I\)  we define \(\operatorname{wt}(\alpha ) = \max \lbrace  \operatorname{wt}I : a_I \ne 0\rbrace \)  and set \(\operatorname{wt}(0):=-\infty \) ; here \(\theta _I\)  denotes the wedge product \(\Lambda _{i\in I}\theta _i\) .
We use primes for the objects associated with \(H^{\prime }\) .
Fix \(p>\nu \) , and let \(f:U\rightarrow H^{\prime }\)  be a \(W^{1,p}_{\operatorname{loc}}\) -mapping for some open subset \(U\subset H\) . If \(\omega \in \Omega ^k(H^{\prime })\)  is a differential \(k\) -form with continuous coefficients, the Pansu pullback of \(\omega \) is the \(k\) -form with measurable coefficients \(f_P^*\omega \)  given by
\(f_P^*\omega (x):=(D_Pf(x))^*\omega (f(x))\,,\) 
where \(D_Pf(x):\mathfrak {h}\rightarrow \mathfrak {h}^{\prime }\)  is the Pansu differential of \(f\)  at \(x\in U\) .
We will use the following special case of [2]}:
Theorem 2.12 (Pullback Theorem (special case))  
Suppose \(\varphi \in C_c^\infty (U)\)  and that \(\alpha \)  and \(\beta \)  are closed left invariant forms
which satisfy
\( \deg \alpha + \deg \beta = N -1 \quad \hbox{and} \quad \operatorname{wt}(\alpha ) + \operatorname{wt}(\beta ) \le -\nu + 1.\) 
Then
\( \int _U f_P^*(\alpha ) \wedge d(\varphi \beta ) = 0.\) 


We now consider Sobolev mappings on the flag manifold.
Let \(f:U\rightarrow  be a continuous map. Then:\begin{itemize}\item f is a {\bf W^{1,p}_{\operatorname{loc}}-mapping} if for every x\in U there is an open neighborhood V of x and group elements g,g^{\prime }\in G=\operatorname{PGL}(n,\mathbb {R}) such that V\subset g\cdot \hat{N}, f(V)\subset g^{\prime }\cdot \hat{N}, and the composition\begin{equation}(\rho (g)\circ \alpha )^{-1}(V)\stackrel{\rho (g)\circ \alpha }{\longrightarrow }V\stackrel{f}{\longrightarrow }g^{\prime }\cdot \hat{N}\stackrel{(\rho (g^{\prime })\circ \alpha )^{-1}}{\longrightarrow }N\end{equation}is a W^{1,p}_{\operatorname{loc}}-mapping.\item The map f is {\bf Pansu differentiable at x\in U} if for some x\in U, V\subset U, g,g^{\prime }\in G as above, the composition (\ref {eqn_f_in_charts}) is Pansu differentiable at (\rho (g)\circ \alpha )^{-1}(x). By the chain rule for Pansu differentials, Pansu differentiability is independent of the choice of g,g^{\prime }, and the resulting Pansu differential \mathfrak {n}\rightarrow \mathfrak {n} is well-defined up to pre/post composition with graded automorphisms. In particular, the property being an isomorphism is well-defined.\end{itemize}Equivalently, one may work directly with the flag manifold as an equiregular subriemannian manifold, and use the notion of Pansu differential in that setting (see \cite [Section 1.4]{gromov_carnot_caratheodory_space_seen_from_within}, \cite {vodopyanov_carnot_manifolds}, and \cite [Appendix A]{KMX1}).\)

Sobolev mappings and the (virtual) preservation of coset foliations
 
The main result in this section is the following:
Lemma 3.1 
Let \(n\ge 4\) .
Let \(U\subset N\)  be a connected open subset, and for some \(p>\nu \)  let \(f:N\supset U\rightarrow N\)  be a \(W^{1,p}_{\operatorname{loc}}\) -mapping whose Pansu differential is an isomorphism almost everywhere. Then after possibly composing with \(\tau \) , if necessary,
for a.e. \(x\in U\)  the Pansu differential \(D_Pf(x)\)  preserves the subspace \(\mathbb {R}X_{i,i+1}\subset V_1\)  for every \(1\le i\le n-1\) .

Corollary 3.2 
Let \(n\ge 4\)  and \(f:U\rightarrow N\)  be as in Lemma REF , and \(K_j\)  be as in Lemma REF (3). Then after possibly composing with \(\tau \) , if necessary, for every \(1\le j\le n-1\) , \(f\)  locally preserves the coset foliation of \(K_j\) , i.e. for every \(x\in U\)  there is an \(r>0\)  such that for every \(g\in N\)  the image of \(gK_j\cap B(x,r)\)  under \(f\)  is contained in a single coset of \(K_j\) .

The corollary follows from Lemma REF  and [4]}. Note that Lemma REF  fails when \(n=3\)  (when \(N\)  is a copy of the Heisenberg group), even for automorphisms. Examples from [5]} show that Lemma REF  can fail even for bilipschitz mappings whose Pansu differential preserves the splitting \(V_1=\mathbb {R}X_{1,2}\oplus \mathbb {R}X_{2,3}\)  for a.e. \(x\) .

We begin with the \(n=4\)  case of the lemma.
As in the proof of Lemma REF , we let
\(\begin{aligned}X_0=X _{23}\,,\;X_1=X _{12}\,,\;X_2=X _{34}\,,\\Y_1=X _{13}\,,\;Y_2=-X _{24}\,,\;Z=X _{14}\,.\end{aligned}\) 
Let \(\alpha _0, \alpha _1, \alpha _2\) , \(\beta _1, \beta _2\) , \(\gamma \)  be the basis
of left invariant 1-forms that are dual to \(X_0, X_1, X_2\) , \(Y_1, Y_2\) , \(Z\) . Then using (REF ) we have
\(d\alpha _0=d\alpha _1=d\alpha _2=0;\\d \beta _1=\alpha _0\wedge \alpha _1,\;\;\; d \beta _2=\alpha _0\wedge \alpha _2;\\d\gamma =\alpha _1\wedge \beta _2+\alpha _2\wedge \beta _1.\) 

Let \(\omega :=\alpha _1\wedge \beta _1\wedge \gamma \) . By Lemma REF (2) on graded automorphisms,
\(f_P^*\omega =u_1\alpha _1\wedge \beta _1\wedge \gamma +u_2\alpha _2\wedge \beta _2\wedge \gamma \) 
where \(u_1\) , \(u_2\)  are measurable and with \(S_i:=\lbrace u_i\ne 0\rbrace \)  the union \(S_1\cup S_2\)  has full measure and \(S_1\cap S_2\)  is null.
Let \(\eta =\alpha _2\wedge \beta _2\)  and pick \(\varphi \in C^\infty _c(U)\) . Then \(\omega \)  and \(\eta \)  are closed left invariant forms and their degrees and weights satisfy the assumption of
the pullback theorem (Theorem REF ), and so
\(\int _U f_P^* \omega \wedge d(\varphi \eta )=0\,.\) 
As \( d(\varphi \eta )=d\varphi \wedge \eta \)  we have
\(f_P^* \omega \wedge d(\varphi \eta )=\pm u_1(X_0\varphi ) d\text{vol}\) 
and therefore \(\int _U u_1(X_0\varphi )d \operatorname{vol}=0\) . Since \(\varphi \)  was arbitrary, it follows that \(X_0u_1=0\)  in the sense of distributions. Similarly by picking \(\eta =\alpha _0\wedge \beta _2\)  we obtain \(X_2u_1=0\) . It follows that \(Y_2u_1=[X_2,X_0]u_1=X_2X_0u_1-X_0X_2u_1=0\) .
Let \(H_i\)  (\(i=1, 2\) ) be the Lie subgroup of \(N\)  whose Lie algebra is generated by
\(X_0\)  and \(X_i\) . Then we see that for almost every left coset \(gH_2\) , \(u_1\)  is locally constant almost everywhere.
Consequently
\(X_0\chi _{S_1}=X_2\chi _{S_1}=Y_2\chi _{S_1}=0\) , where \(\chi _S\)  denotes the characteristic function of a subset \(S\subset U\) .
Similarly by using \(\eta =\alpha _1\wedge \beta _1\) , \(\alpha _0\wedge \beta _1\) , we obtain
\(X_0u_2=X_1u_2=Y_1u_2=0\)  and \(X_0\chi _{S_2}=X_1\chi _{S_2}=Y_1\chi _{S_2}=0\) . As \(\chi _{S_1}+\chi _{S_2}=1\) , we infer that \(X_i\chi _{S_j}=0\)  for all
\(0\le i\le 2\)  and \(j=1, 2\) . Since \(X_0, X_1, X_2\)  generates \(\mathfrak {n}\) , this gives \(X\chi _{S_j}=0\)  for every \(X\in \mathfrak {n}\) , and hence \(\chi _{S_j}\)  is locally constant a.e. By the connectedness of \(U\) , \(\chi _{S_j}\)  is constant a.e.
As \(\chi _{S_1}\)  takes on only two possible values (0 or 1), we have
\(\chi _{S_1}=0\)  a.e. or \(\chi _{S_1}=1\)  a.e.
If \(\chi _{S_1}=1\)  a.e., then by the definition of \(S_1\)  we have
\(Df_x(\mathbb {R} X_{i})=\mathbb {R} X_{i}\)  for all \(0\le i\le 2\)  and a.e. \(x\in U\) . If
\(\chi _{S_1}=0\)  a.e., then
\(\tau \circ Df_x(\mathbb {R} X_{i})=\mathbb {R} X_{i}\)  for a.e. \(x\in U\) .
Now assume \(n\ge 5\) .
Let \(\lbrace \theta _{i,j}\mid 1\le i<j\le n\rbrace \)  be the basis of left invariant 1-forms on \(N\)  that are dual to the basis \(\lbrace X_{i,j}\mid 1\le i<j\le n\rbrace \)  of \(\mathfrak {n}\) . Then we have
\(d \theta _{i,j}=\left\lbrace \begin{array}{rl}0& \text{ if } j=i+1\\-\sum _{k=i+1}^{j-1}\theta _{i, k}\wedge \theta _{k,j} & \text{ if } j>i+1\end{array}\right.\) 
Let \(\omega _+=\theta _{1,2}\wedge \theta _{1,3}\wedge \cdots \wedge \theta _{1,n}\)  and
\(\omega _-=\theta _{n-1,n}\wedge \theta _{n-2,n}\wedge \cdots \wedge \theta _{1,n}\) . Note that
\(\omega _+\)  is closed since by (REF ) we have \(d\theta _{1,2}=0\) , and
\(d\theta _{1,j}=-\sum _{k=2}^{j-1}\theta _{1,k}\wedge \theta _{k,j}\)  for \(j\ge 3\)  and so
\(\theta _{1,2} \wedge \cdots \wedge \theta _{1, j-1}\wedge d\theta _{1,j}=0\) . Similarly
\(\omega _-\)  is closed. By Lemma REF  we have
\(f^*_P(\omega _+)=u_+\omega _++u_-\omega _-\) 
for some measurable functions \(u_+\) , \(u_-\) .
For \(2\le k\le n-1\) , let
\(\eta _{k-}=\bigwedge _{2\le i<j\le n, (i,j)\ne (k, k+1)} \theta _{i,j},\) 
and for \(1\le k\le n-2\) , let
\(\eta _{k+}=\bigwedge _{1\le i<j\le n-1, (i,j)\ne (k, k+1)} \theta _{i,j}.\) 
We show that \(\eta _{k-}\)  is closed. First we have \(d\theta _{l, l+1}=0\) . For \(m>l+1\) ,
\(d\theta _{l, m}=-\sum _{s=l+1}^{m-1}\theta _{l, s}\wedge \theta _{s, m}\) 
and at least one of the two terms \(\theta _{l, s}\) , \(\theta _{s, m}\)  is already present in
\(\bigwedge _{2\le i<j\le n, (i,j)\ne (k, k+1), (l,m)} \theta _{i,j};\) 
it follows that
\(d\theta _{l, m}\wedge \bigwedge _{2\le i<j\le n, (i,j)\ne (k, k+1), (l,m)} \theta _{i,j}=0\) .
Similarly \(\eta _{k+}\)  is closed.
We next show that the degree and weight conditions in the pullback theorem (Theorem REF ) are satisfied: for \(\omega _+\)  and \(\eta =\eta _{k-}\)  or \(\eta _{k+}\)  we have \(\text{degree}(\omega _+)+\text{degree}(\eta )=N-1\)  and \(\text{wt}(\omega _+)+\text{wt}(\eta )=-\nu +1\) .
First these conditions are satisfied by \(\omega _+\)  and \(\eta _-\)  as
\(\omega _+\wedge \eta _-=\pm \bigwedge _{(i,j)\ne {k, k+1}}\theta _{ij}\)  and
\(\theta _{k, k+1}\)  has degree one and weight \(-1\) . These conditions are also satisfied by \(\omega _+\)  and \(\eta _+\)  as \(\tau (\eta _-)=\pm \eta _+\)  and so \(\eta _-\) , \(\eta _+\)  have the same degree and weight.
Hence by Theorem REF , for any \(\varphi \in C^\infty _c(U)\) , we have \(\int _U f_P^*(\omega _+)\wedge d(\varphi \eta )=0\) .
As \(d(\varphi \eta )=d\varphi \wedge \eta \) ,
we get \(f_P^*(\omega _+)\wedge d(\varphi \eta _{k-})=\pm u_+X_{k,k+1}\varphi \text{Vol}\)  (for \(2\le k\le n-1\) ) and
\(f_P^*(\omega _+)\wedge d(\varphi \eta _{k+})=\pm u_-X_{k,k+1}\varphi \text{Vol}\)  (for \(1\le k\le n-2\) ).
It follows that distributionally we have
\(X_{k,k+1} u_+=0\)  for all \(2\le k\le n-1\)  and
\(X_{k,k+1} u_-=0\)  for all \(1\le k\le n-2\) .
Let \(S_+\subset U\)  be the subset where the Pansu differential of \(f\)  fixes all the directions \(\mathbb {R}X_{i, i+1}\) .
Similarly let \(S_-\subset U\)  be the set of points \(x\in U\)  such that
\(\tau \circ Df_x\)  fixes all the directions \(\mathbb {R}X_{i, i+1}\) . Again by Lemma REF  (2),
\(S_+\cup S_-\)  has full measure in \(U\)  and \(S_+\cap S_-\)  is null.
Notice that
\(S_+\)  agrees with \(\lbrace x\in U\mid u_+(x)\ne 0\rbrace \)  up to a set of measure 0 and \(S_-\)  agrees with \(\lbrace x\in U\mid u_-(x)\ne 0\rbrace \)  up to a set of measure 0. Hence \(X_{k,k+1} \chi _{S_+}=0\)  for all \(2\le k\le n-1\)  and \(X_{k,k+1} \chi _{S_-}=0\)  for all \(1\le k\le n-2\) .
Since \(\chi _{S_+}+\chi _{S_-}=1\)  we have \(X_{k,k+1} \chi _{S_+}=0\)  for all \(1\le k\le n-2\) .
So \(X_{k,k+1} \chi _{S_+}=0\)  for all \(1\le k\le n-1\) . Since the \(X_{k, k+1}\) , \(1\le k\le n-1\)  generates the Lie algebra \(\mathfrak {n}\) , we see that \(X \chi _{S_+}=0\)  for any \(X\in \mathfrak {n}\)  and we conclude as in the \(n=4\)  case.


Rigidity for fibration preserving maps
In the section we show that mappings of \( which respect the fibrations \) j:j\( for all \) 1jn-1\( are locally projective.\)
Definition 4.1 A mapping \(f:U\rightarrow  is {\bf fibration-preserving} if for every \) 1jn-1\(, and every \) Wjj\(, the image of \) Uj-1(Wj)\( under \) f\( is contained in \) j-1(Wj')\( for some \) Wj'j\(. Equivalently, letting \) Uj:=j(U)\(, there exist mappings \) fj:Ujj\( such that \) jf=fjj\(. A mapping \) f:U is locally fibration preserving if it is fibration preserving near any point in \(U\) .

Proposition 4.2  
Suppose \(n\ge 3\)  and \(f:U\rightarrow  is a continuous, locally fiber preserving mapping, where \) U is a connected open subset. If for some \(x\in U\)  the Pansu differential \(D_Pf(x)\)  exists and is an isomorphism (see Subsection REF ) then \(f\)  agrees with \(g: for some \) gPGL(n,R)\(.\)

Proof of Proposition REF  in the \(n=3\)  case
In this subsection we fix \(n=3\) , and without further mention \( and \) i\(, and \) i:i\( will be the objects associated with \) R3\(.\)
Definition 4.3 A projective frame is an indexed collection \(\lbrace W_1^i\rbrace _{0\le i\le 3}\subset 1\)  where any three elements span \(\mathbb {R}^3\) . The standard projective frame \(\lbrace \hat{W}_1^i\rbrace _{0\le i\le 3}\)  is given by \(\hat{W}_1^i=\operatorname{span}(e_i)\)  for \(1\le i\le 3\) , and \(\hat{W}_1^0=\operatorname{span}(e_1+e_2+e_3)\) .

Note that for any projective frame \(\lbrace W_1^i\rbrace _{0\le i\le 3}\subset 1\)  there exists a unique \(g\in \operatorname{PGL}(3,\mathbb {R})\)  such that \(g(\hat{W}_1^i)=\hat{W}_1^i\)  for \(0\le i\le 3\) , i.e. \(\operatorname{PGL}(3,\mathbb {R})\)  acts freely transitively on the collection of projective frames.
The proof of Proposition REF  is inspired by the following elementary classical result:
Theorem 4.4 (Fundamental Theorem of Projective Geometry)  

Every fibration-preserving bijection \(f: agrees with some \) gPGL(3,R)\(.\) 
Note that this is an equivalent reformulation of the classical result using the flag manifold rather the (traditional) projective plane.
Since \(f\)  is a bijection, so are the maps \(f_i:i\rightarrow i\)  for \(i\in \lbrace 1,2\rbrace \) . It follows that \(f_1\)  maps \(\pi _1(\pi _2^{-1}(W_2))\)  bijectively to \(\pi _1(\pi _2^{-1}(f_2(W_2)))\)  for every plane \(W_2\in 2\) . Therefore three elements of 1 lie in a plane if and only if their images under \(f_1\)  lie in a plane. Hence \(\lbrace f_1(\hat{W}_1^i)\rbrace _{0\le i\le 3}\)  is a projective frame, so after composing \(f\)  with some element of \(\operatorname{PGL}(3,\mathbb {R})\) , we may assume that \(f_1(\hat{W}_1^i)=\hat{W}_1^i\)  for \(0\le i\le 3\) . It follows that \(f_1\)  also fixes \(\hat{W}_1^4:=\hat{W}_2^{30}\cap \hat{W}_2^{12}\) .
We identify \(\mathbb {R}^2\)  with the affine plane \(\lbrace W_1\in 1\mid W_1\lnot \subset \hat{W}_2^{12}=\operatorname{span}(e_1,e_2)\rbrace \)  by \((x_1,x_2)\leftrightarrow \operatorname{span}(x_1,x_2,1)\) . We may define a bijection \(\phi :\mathbb {R}^2\rightarrow \mathbb {R}^2\)  by \(\operatorname{span}(\phi (x_1,x_2),1)=f_1(\operatorname{span}(x_1,x_2,1))\) . Using the fact that \(f_1(\hat{W}_1^i)=\hat{W}_1^i\)  for \(0\le i\le 4\) , one sees that:

\(\phi \)  fixes \((0,0)\) , \((1,1)\) .

\(\phi \)  maps lines bijectively to lines.

For every \(v\in \lbrace e_1,e_2,e_1+e_2\rbrace \)  and every line \(L\)  parallel to \(v\) , the image \(f(L)\)  is a line parallel to \(v\) .

Using a geometric construction (see Figure REF ) it follows that the restriction of \(\phi \)  to \(\mathbb {R}\times \lbrace 0\rbrace \simeq \mathbb {R}\)  is a field isomorphism. Since \(\operatorname{id}_{\mathbb {R}}\)  is the only automorphism of \(\mathbb {R}\) , it follows that \(\phi \)  fixes \(\mathbb {R}\times \lbrace 0\rbrace \) . Hence it also fixes \(\lbrace 0\rbrace \times \mathbb {R}\) , then all of \(\mathbb {R}^2\) ; and \(f_1\)  fixes 1, and \(f\)  fixes \(.\)
<FIGURE>The \(n=3\)  case of Proposition REF  may be viewed as a localized version of Theorem REF  for mappings which are continuous, but not necessarily bijective.
Before proceeding, we first give a rough idea how the argument goes.
The first step is to show that local projectivity of \(f\)  propagates: if \(f\)  locally agrees with some \(g\in \operatorname{PGL}(3,\mathbb {R})\)  near some point \(x\in U\) , then \(f\)  will locally agree with \(g\)  near points in the fibers of \(\pi _1\) , \(\pi _2\)  passing through \(x\)  (see Lemma REF ). This readily implies that the subset of \(U\)  where \(f\)  agrees with \(g\)  is a connected component of \(U\) . Thus we are reduced to finding a single point near which \(f\)  is locally projective. To that end, we consider \((W_1,W_2)\in U\)  a point of differentiability where differential is an isomorphism. Using the definition of differentiability, we argue that the map \(f\)  is nondegenerate near \((W_1,W_2)\) , in the sense that the map \(f_1\)  induced on 1 carries a projective frame localized near \(W_1\)  to a projective frame (see below the claim in the proof of the \(n=3\)  case of Proposition REF ). Then by pre/post-composing with suitable elements of \(\operatorname{PGL}(3,\mathbb {R})\) , and working in 1, we are able to reduce to a map \(\phi :\mathbb {R}^2\supset U\rightarrow \mathbb {R}^2\)  which satisfies a localized version of the conditions (a)-(c) appearing in the proof of Theorem REF ; this is shown in Lemma REF  to be locally affine, which then implies that \(f\)  is locally projective near \((W_1,W_2)\) .
We now return to the preparations for the proof of Proposition REF .

Lemma 4.5 (Propagation of projectivity) 

Suppose \(U\subset  is open and \) f:U is continuous and fibration preserving.

If \(f\)  agrees with \(g\in \operatorname{PGL}(3,\mathbb {R})\)  near \((\bar{W}_1,\bar{W}_2)\in U\) , then \(f\)  agrees with \(g\)  near the fiber \(\pi _i^{-1}(\bar{W}_i)\cap U\) , for \(i\in \lbrace 1,2\rbrace \) .

For every \(g\in \operatorname{PGL}(3,\mathbb {R})\) , the set where \(f\)  locally agrees with \(g\)  is a connected component of \(U\) .


(1). We prove the assertion for \(i=2\) ; the case when \(i=1\)  is similar.
After postcomposing with an element of \(\operatorname{PGL}(3,\mathbb {R})\) , may assume without loss of generality that \(f\equiv \operatorname{id}\)  on an open set \(V\subset U\)  with \((\bar{W}_1,\bar{W}_2)\in V\) .
Pick \((W_1,W_2)\in U\cap \pi _2^{-1}(\bar{W}_2)\) , i.e. \(W_2=\bar{W}_2\) . Choose \((W_1,W_2^{\prime })\in U\setminus \lbrace (W_1,W_2)\rbrace \)  such that \(\pi _2^{-1}(W_2^{\prime })\cap V\ne \emptyset \) .
Now suppose \((W_1^j,W_2^j)\rightarrow (W_1,W_2)\) . We may choose \(\lbrace W_2^{\prime j}\rbrace \subset 2\)  such that \(W_1^j\subset W_2^{\prime j}\)  and \(W_2^{\prime j}\rightarrow W_2^{\prime }\)  as \(j\rightarrow \infty \) . Since \(V\)  is open, after dropping finitely many terms, we may assume that \(\pi _2^{-1}(W_2^j)\cap V\ne \emptyset \)  and \(\pi ^{-1}(W_2^{\prime j})\cap V\ne \emptyset \) . Since \(f\)  is fibration-preserving and \(f\equiv \operatorname{id}\)  on \(V\) , it follows that \(f(\pi _2^{-1}(W_2^j)\cap U)\subset \pi _2^{-1}(W_2^j)\)  and \(f(\pi ^{-1}(W_2^{\prime j})\cap U)\subset \pi ^{-1}(W_2^{\prime j})\) . The map \(f\)  is fibration preserving, so \(f(W_1^j,W_2^j), f(W_1^j,W_2^{\prime j})\in \pi _1^{-1}(f_1(W_1^j))\) . Therefore \(\pi _1^{-1}(f_1(W_1^j))\)  intersects both \(\pi _2^{-1}(W_2^j)\)  and \(\pi _2^{-1}(W_2^{\prime j})\)  nontrivially, forcing \(f_1(W_1^j)=W_2^j\cap W_2^{\prime j}=W_1^j\) . Thus \(f(W_1^j,W_2^j)=(W_1^j,W_2^j)\) . Since the sequence \(\lbrace (W_1^j,W_2^j)\rbrace \)  was arbitrary, this proves (1).
(2). Define an equivalence relation on \(U\)  where \(x,x^{\prime }\in U\)  are equivalent if there is a path \(\gamma :[0,1]\rightarrow U\)  from \(x\)  to \(x^{\prime }\)  which is piecewise contained in a fiber of one of the fibrations \(\pi _i:i\) . We claim that the equivalences classes are open subsets of \(U\) . To see this, pick \(x=g\cdot (W_1^-,W_2^-)\in U\) , and note that the vector fields \(X_{1,2}\) , \(X_{2,3}\)  on \(N\)  are bracket generating and tangent to the cosets of \(K_1\)  and \(K_2\) , respectively; and hence their pushforwards \(X_{1,2}^{\prime }\) , \(X_{2,3}^{\prime }\)  under the composition \(N\stackrel{\alpha }{\rightarrow }\hat{N}\stackrel{g}{\rightarrow }g\cdot \hat{N}\)  are bracket generating vector fields on \(g\cdot \hat{N}\)  which are tangent to the fibers of \(\pi _1\)  and \(\pi _2\)  respectively, by Lemma REF (4). By [6]}
the equivalence class of \(x\)  contains a neighborhood of \(x\) . It follows that the equivalence classes are connected components of \(U\) . Since the set where \(f\)  locally agrees with \(g\)  is a union of equivalences classes by (1), assertion (2) follows.
Lemma 4.6 
Suppose \(U\subset \mathbb {R}^2\)  is a connected open subset, and
\(\phi =(\phi _1,\phi _2):\mathbb {R}^2\supset U\rightarrow \mathbb {R}^2\) 
is a continuous map such for every \(v\in \lbrace e_1,e_2,e_1+e_2\rbrace \)  and every line \(L\)  parallel to \(v\) , the image of \(L\cap U\)  is contained in a line parallel to \(v\) .
Then \(\phi \)  is of the form
\(\phi (x,y)=(mx+b_1,my+b_2)\) 
for some \(m,b_1,b_2\in \mathbb {R}\) .

First assume that \(\phi \)  is smooth.
Applying the hypothesis with \(v\in \lbrace e_1,e_2\rbrace \)  implies that \(2\phi _1=1\phi _2\equiv 0\) , so \(\phi _i\)  depends only on \(x_i\) .
Applying the hypothesis when \(v=e_1+e_2\)  we have
\(0\equiv (1+2)(\phi _2-\phi _1)=2\phi _2-1\phi _1\,.\) 
But since \(i\phi _i\)  depends only on \(x_i\)  this forces \(1\phi _1=2\phi _2=const\)  and so (REF ) holds. Since the conditions are preserved by taking linear combinations and precomposing with translations, the general case follows by mollification.
Alternatively, one may argue as follows. Without loss of generality, one may assume that \(\phi (0,0)=(0,0)\) . By geometric contruction, \(\phi (x+x^{\prime },0)=\phi (x,0)+\phi (x^{\prime },0)\)  when \(x,x^{\prime }\in (-r,r)\)  for \(r\)  small. Hence \(\phi (x_1,0)=mx_1\)  for some \(m\in \mathbb {R}\) , for \(x_1\in (-r,r)\) . Invoking the hypotheses again, we get \(\phi (x_1,x_2)=(mx_1,mx_2)\) . Thus the lemma holds locally, i.e. for every \((x_1,x_2)\in U\)  there is an open set \(V_{x_1,x_2}\)  containing \((x_1,x_2)\)  and \(m,b_1,b_2\)  depending on \((x_1,x_2)\)  such that (REF ) holds in \(V_{x_1,x_2}\) ; since \(U\)  is connected and \(m,b_1,b_2\)  are locally constant, it follows that they are independent of \(x_1,x_2\) , and so (REF ) holds.

Definition 4.8 An indexed tuple \(\lbrace W_1^i\rbrace _{0\le i\le 4}\subset 1\)  is an augmented projective frame if \(\lbrace W_1^i\rbrace _{0\le i\le 3}\)  is a projective frame and
\(W_1^4=\operatorname{span}(W_1^3,W_1^0)\cap \operatorname{span}(W_1^1,W_1^2)\,.\) 
The standard augmented projective frame \(\lbrace \hat{W}_1^i\rbrace _{0\le i\le 4}\)  is given by
\(\hat{W}_1^i={\left\lbrace \begin{array}{ll}e_1+e_2+e_3,\quad &i=0\\e_i,\quad &1\le i\le 3\\e_1+e_2,\quad &i=4\end{array}\right.}\) 


Given a subset \(\Sigma \subset 1\) , we obtain (possibly empty) subsets of 2 and \(:\begin{equation}\begin{aligned}2(\Sigma )&:=\lbrace \operatorname{span}(\sigma _1,\sigma _2)\mid \; \sigma _i\in \Sigma \,,\;\sigma _1\ne \sigma _2\rbrace \,.\\\Sigma )&:=\lbrace (W_1,W_2)\mid W_1\in \Sigma \,,\;W_2\in 2(\Sigma )\rbrace \,.\end{aligned}\end{equation}\)
Lemma 4.10 
There is an augmented projective frame \(\lbrace \tilde{W}_1^i\rbrace _{0\le i\le 4}\subset 1\)  such that \(\lbrace \tilde{W}_1^i\rbrace _{0\le i\le 4})\subset \hat{N}\)  and
\((\tilde{W}_1^3,\operatorname{span}(\tilde{W}_1^3,\tilde{W}_1^2))=(W_1^-,W_2^-)\,.\) 


Since \(\hat{N}\)  is an open dense subset of \( by Lemma~\ref {lem_properties_of_f}, we may choose \) gG\( such that \)\(g\cdot \lbrace \hat{W}_1^i\rbrace _{0\le i\le 4})=\lbrace g\cdot \hat{W}_1^i\rbrace _{0\le i\le 4})\)\( lies in \) N\(. Then we may let \) W1i:=(ng)W1i\( for \) 0i4\(, where \) nN\( satisfies \) n(g(W13,span(W13,W12)))=(W1-,W2-)\(.\)

By Lemma REF (2) and the connectedness of \(U\) , it suffices to show that \(f\)  locally agrees with some element of \(\operatorname{PGL}(3,\mathbb {R})\)  near \(x\) . Therefore after shrinking \(U\)  we may assume that \(f\)  is fibration-preserving, not just locally fibration-preserving.
After pre/postcomposing with elements of \(\operatorname{PGL}(3,\mathbb {R})\) , we assume without loss of generality that \(x=(W_1^-,W_2^-)\in U\) , \(f(x)=x\) , and \(D_Pf(x)\)  is defined and an isomorphism.
Let \(\lbrace \tilde{W}_1^i\rbrace _{0\le i\le 4}\subset \hat{N}\)  be the augmented projective frame from Lemma REF . For \(r>0\)  let \(\lbrace W_1^i\rbrace _{0\le i\le 4}\)  be the image of \(\lbrace \tilde{W}_1^i\rbrace _{0\le i\le 4}\)  under \(\hat{\delta }_r\) ; we may assume \(r\)  is small enough that \(\lbrace W_1^i\rbrace _{0\le i\le 4})\in U\) .
Claim If \(r\)  is sufficiently small, then \(\lbrace f_1(W_1^i)\rbrace _{0\le i\le 3}\)  is a projective frame.

 
Since \(f\)  is Pansu differentiable at \(x\)  and \(D_Pf(x)\)  is an isomorphism
\(\hat{\delta }_{r^{-1}}\circ f\circ \hat{\delta }_r\rightarrow D_Pf(x)\) 
uniformly on compact sets as \(r\rightarrow 0\) , where \(D_Pf:\hat{N}\rightarrow \hat{N}\)  is a graded automorphism; here we identify \(N\)  with \(\hat{N}\) .
Note that \(D_Pf(x):\hat{N}\rightarrow \hat{N}\)  is fibration-preserving because it is a limit of fibration-preserving maps. By Lemma REF (3) there exist \(\lambda _1,\lambda _2,\lambda _3\ne 0\)  such that \(\hat{\Phi }=\operatorname{diag}(\lambda _1,\lambda _2,\lambda _3) \in \operatorname{PGL}(3,\mathbb {R})\)  agrees with \(D_Pf(x)\)  on \(\hat{N}\) . Therefore \((\hat{\Phi })_1\)  maps \(\lbrace \tilde{W}_1^i\rbrace _{0\le i\le 3}\)  to a projective frame.
It follows that for \(r\)  small both \(\lbrace (\hat{\delta }_{r^{-1}}\circ f\circ \hat{\delta }_r)_1(\tilde{W}_1^i)\rbrace _{0\le i\le 3}\)  and \(\lbrace f_1(W_1^i)=(f\circ \hat{\delta }_r)_1(\tilde{W}_1^i)\rbrace _{0\le i\le 3}\)  are projective frames.
Let \(\lbrace \hat{W}_1^i\rbrace _{0\le i\le 4}\)  be the standard augmented projective frame, and let \(\hat{W}_2^{ij}:=\operatorname{span}(\hat{W}_1^i,\hat{W}_1^j)\)  for \(0\le i\ne j\le 4\) .
Since \(\lbrace W_1^i\rbrace _{0\le i\le 3}\)  and \(\lbrace f_1(W_1^i)\rbrace _{0\le i\le 3}\)  are both projective frames, there are elements \(g_1,g_2\in \operatorname{PGL}(3,\mathbb {R})\)  such that \(g_1(\hat{W}_1^i)=W_1^i\)  and \(g_2(f_1(W_1^i))=\hat{W}_1^i\)  for \(0\le i\le 3\) . We let \(\hat{U}:=g_1^{-1}(U)\subset  and \) f:=g2fg1:U.
Then \(\lbrace \hat{W}_1^i\rbrace _{0\le i\le 4})\subset \hat{U}\) , the map \(\hat{f}\)  is fibration-preserving, and \((\hat{f})_1(\hat{W}_1^i)=\hat{W}_1^i\)  for all \(0\le i\le 3\) . Since two distinct lines lie in a unique plane, and two distinct planes intersect in a line, the fact that \(\hat{f}\)  is fibration-preserving implies that:

\(\hat{f}_2(\hat{W}_2^{ij})=\hat{W}_2^{ij}\)  for \(0\le i\ne j\le 3\) .

\(\hat{f}_1(\hat{W}_1^4)=(\hat{f})_1(\hat{W}_2^{30}\cap \hat{W}_2^{12})=\hat{W}_1^4\) .

\(\hat{f}_2(\hat{W}_2^{ij})=\hat{W}_2^{ij}\)  for \(0\le i\ne j\le 4\) .

\(\hat{f}(\hat{W}_1^i,\hat{W}_2^{ij})=(\hat{W}_1^i,\hat{W}_2^{ij})\)  for \(0\le i\ne j\le 4\) .

Since \(\hat{f}_1([e_3])=[e_3]\) ,
for small \(r>0\)  we may define \(\phi :\mathbb {R}^2\supset B(0,r)\rightarrow \mathbb {R}^2\)  by \(\operatorname{span}(\phi (x_1,x_2),1)=\hat{f}_1(\operatorname{span}(x_1,x_2,1))\) .
The map \(\hat{f}\)  is fibration-preserving and it fixes the standard augmented projective frame, so the hypotheses of Lemma REF  hold for \(\phi \) . Applying Lemma REF  to \(\phi \) , we get that for some \(m\in \mathbb {R}\)  we have \(\hat{f}_1([x_1,x_2,1])=[mx_1,mx_2,1]\)  for \(x_1,x_2\in \mathbb {R}\)  small.
Suppose \(m=0\) . Then \(\hat{f}\) , and hence also \(f\) , takes values in a single fiber of \(\pi _1\)  near \(x=(W_1^-,W_2^-)\) . It follows that the Pansu differential \(D_Pf(x)=\lim _{r\rightarrow 0}\hat{\delta }_{r^{-1}}\circ f\circ \delta _r\)  takes values in a single fiber of \(\pi _1\) ; this contradicts the nondegeneracy of the Pansu differential. Hence \(m\ne 0\) .
Let \(\hat{g}:=\operatorname{diag}(m,m,1)\in \operatorname{PGL}(3,\mathbb {R})\) . Then \(\hat{g}_1:1\rightarrow 1\)  agrees with \(\hat{f}_1\)  near \([e_3]\) . Since \(\hat{g}\)  and \(\hat{f}\)  are both fibration-preserving, they agree near \(x=(W_1^-,W_2^-)\) .

Proof of Proposition REF , general case
The \(n\ge 4\)  case is similar to the \(n=3\)  case. The replacement for Lemma REF  is:
Lemma 4.12 
Suppose \(V\subset \mathbb {R}^n\)  is a connected open subset, and
\(\phi =(\phi _1,\ldots ,\phi _n):V\rightarrow \mathbb {R}^n\) 
is a continuous map such for every \(v\in \lbrace e_1,\ldots ,e_n,e_1+\ldots +e_n\rbrace \)  and every line \(L\)  parallel to \(v\) , the image \(\phi (L\cap V)\)  is contained in a line parallel to \(v\) .
Then \(\phi \)  is of the form
\(\phi (x_1,\ldots ,x_n)=(mx_1+b_1,\ldots ,mx_n+b_n)\) 
for some \(m,b_1,\ldots ,b_n\in \mathbb {R}\) .

We omit the proof as it is similar to the proof of Lemma REF .
The \(n\ge 4\)  version of Lemma REF  is:
Lemma 4.14 
Suppose \(U\subset  is open and \) f:U is fibration preserving.

Suppose \(f\)  agrees with \(g\in \operatorname{PGL}(n,\mathbb {R})\)  near \((\bar{W}_1,\ldots ,\bar{W}_{n-1})\in U\) . For \(i\in \lbrace 1, \cdots , n-1\rbrace \) , let \(V_i\)  be the connected component of \(\pi _i^{-1}(\bar{W}_i)\cap U\)  containing \((\bar{W}_1,\ldots ,\bar{W}_{n-1})\) . Then \(f\)  agrees with \(g\)  near \(V_i\) .

For every \(g\in \operatorname{PGL}(n,\mathbb {R})\) , the set where \(f\)  locally agrees with \(g\)  is a connected component of \(U\) .


We prove the lemma by induction on the dimension \(n\ge 3\) . Lemma REF  covers the case \(n=3\) , so we may assume inductively that the lemma holds for dimensions strictly smaller than \(n\) .
(1). We may assume without loss of generality that \(g=\operatorname{id}\) . Suppose \(f\equiv \operatorname{id}\)  on an open subset \(V\subset U\)  containing \((\bar{W}_1,\ldots ,\bar{W}_{n-1})\) . Arguing by contradiction, suppose \(i\in \lbrace 1, \cdots , n-1\rbrace \) , and for some \((W_1,\ldots ,W_{n-1})\in V_i\) , there is a sequence \( \lbrace (W_1^j,\ldots ,W_{n-1}^j)\rbrace \subset U\)  which converges to \((W_1,\ldots ,W_{n-1})\)  as \(j\rightarrow \infty \) , but
\(f(W_1^j,\ldots ,W_{n-1}^j)\ne (W_1^j,\ldots ,W_{n-1}^j)\) 
for all \(j\) . After passing to a subsequence, we may assume that for all \(j\)  the connected component of \(\pi _i^{-1}(W_i^j)\cap U\)  containing \((W_1^j,\ldots ,W_{n-1}^j)\)  intersects \(V\) . Since \(f\)  is fibration preserving and \(f\equiv \operatorname{id}\)  on \(V\) , it follows that \(f\)  maps \(\pi _i^{-1}(W_i^j)\cap U\)  into \(\pi _i^{-1}(W_i^j)\) . Identifying \(\pi _i^{-1}(W_i^j)\)  with the flag manifold in \(\mathbb {R}^{n-1}\) , the restriction \(f\)  to \(\pi _i^{-1}(W_i^j)\)  induces a fibration-preserving mapping; by the induction assumption, since \(f\)  fixes \(V\)  it will also fix \((W_1^j,\ldots ,W_{n-1}^j)\) . This is a contradiction. Hence (1) holds.
(2). Note that each element of the basis \(X_{1,2},\ldots ,X_{n-1,n}\)  for \(V_1\subset \mathfrak {n}\)  is tangent to one of the subgroups \(K_j\)  for \(1\le j\le n-1\) . Hence we may use Lemma REF (4) and argue as in Lemma REF (2).
Definition 4.15 An indexed tuple \(\lbrace W_1^i\rbrace _{0\le i\le n}\subset 1\)  is a projective frame if any subset of \(n\)  elements spans \(\mathbb {R}^n\) . The standard projective frame \(\lbrace \hat{W}_1^i\rbrace _{0\le i\le n}\)  is given by \(\hat{W}_1^i=e_i\)  for \(1\le i\le n\) 
and \(\hat{W}_1^0=e_1+\ldots +e_n\) .
An indexed tuple \(\lbrace W_1^i\rbrace _{0\le i\le n+1}\subset 1\)  is an augmented projective frame if \(\lbrace W_1^i\rbrace _{0\le i\le n}\)  is a projective frame and \(W_1^{n+1}=\operatorname{span}(W_1^n,W_1^0)\cap \operatorname{span}(W_1^1,\ldots ,W_1^{n-1})\) . The standard augmented projective frame is \(\lbrace \hat{W}_1^i\rbrace _{0\le i\le n+1}\)  with \(\hat{W}_1^{n+1}=e_1+\ldots +e_{n-1}\) .

Given a subset \(\Sigma \subset 1\) , we obtain (possibly empty) subsets of \(j\)  and \(:{\begin{@align*}{1}{-1}j(\Sigma )&:=\lbrace \operatorname{span}(\Sigma ^{\prime })\mid \; \Sigma ^{\prime }\subset \Sigma ,\; |\Sigma ^{\prime }|=j,\; \dim \operatorname{span}(\Sigma ^{\prime })=j\rbrace \,.\\&\Sigma ):=\lbrace (W_1,\ldots ,W_{n-1})\in W_j\in j(\Sigma )\rbrace \,.\end{@align*}}\)
Lemma 4.16 There is an augmented projective frame \(\lbrace \tilde{W}_1^i\rbrace _{0\le i\le n+1}\subset 1\)  such that:

\(\lbrace \tilde{W}_1^i\rbrace _{0\le i\le n+1})\)  is contained in \(\hat{N}\) .

\(\operatorname{span}(\tilde{W}_1^n,\ldots ,\tilde{W}_1^{n-j+1}) =\operatorname{span}(e_n,\ldots ,e_{n-j+1})\)  for all \(1\le j\le n-1\) .


This follows as in the proof of Lemma REF .

The proof parallels the \(n=3\)  case closely, so we will be brief.
It suffices to show that \(f\)  locally agrees with some element of \(\operatorname{PGL}(n,\mathbb {R})\)  near \(x\) . Also, we may assume without loss of generality that \(f\)  is fibration-preserving, \(x=(W_1^-,\ldots ,W_{n-1}^-)\in U\) , \(f(x)=x\) , and that \(D_Pf(x)\)  is well-defined and an isomorphism.
For \(r>0\)  let \(W_1^i:=\hat{\delta }_r(\tilde{W}_1^i)\)  for \(0\le i\le n+1\) ; we take \(r\)  small enough that \(\lbrace W_1^i\rbrace _{0\le i\le n+1})\subset U\) .
Claim For \(r\)  small \(\lbrace f_1(W_1^i)\rbrace _{0\le i\le n}\subset 1\)  is a projective frame.

We omit the proof, as it is similar to claim in the proof of the \(n=3\)  case.
Let \(g_1,g_2\in \operatorname{PGL}(n,\mathbb {R})\)  be such that \(g_1(\hat{W}_1^i)=W_1^i\) , \(g_2(f_1(W_1^i))=\hat{W}_1^i\)  for \(0\le i\le n\) . We now define \(\hat{U}=g_1^{-1}(U)\)  and \(\hat{f}:=g_2\circ f\circ g_1:\hat{U}\rightarrow .\) Arguing as in the \(n=3\)  case, one obtains that \(\hat{f}\)  is fibration-preserving, \( \lbrace \hat{W}_1^i\rbrace _{0\le i\le n+1})\subset \hat{U}\) ,
\(f_j\)  fixes \(j(\lbrace \hat{W}_1^i\rbrace _{0\le i\le n+1})\)  elementwise and \(f\)  fixes \( \lbrace \hat{W}_1^i\rbrace _{0\le i\le n+1})\)  elementwise.
For \(r>0\)  small we define \(\phi :\mathbb {R}^n\supset B(0,r)\rightarrow \mathbb {R}^n\)  by
\(\operatorname{span}(\phi (x_1,\ldots ,n),1)=f_1(\operatorname{span}(x_1,\ldots ,x_n,1))\,.\) 
Applying Lemma REF , for some \(m\in \mathbb {R}\)  we get
\(\phi (x_1,\ldots ,x_n)=(mx_1,\ldots ,mx_n)\,.\) 
As in the \(m=3\)  case we see that \(m\ne 0\) , and that \(f_1\)  agrees with \(g:=\operatorname{diag}(m,\ldots ,m,1)\)  near \(e_n\) . This implies that \(f\)  agrees with \(g\)  near \(x=(W_1^-,\ldots ,W_{n-1}^-)\) .


The proof of Theorem  REF
For every \(x\in U\)  choose a connected open set \(U_x\subset U\)  containing \(x\)  and group elements \(\bar{g}_x,\bar{g}_x^{\prime }\in \operatorname{PGL}(n,\mathbb {R})\)  such that \(\bar{g}_x(U_x),\bar{g}_x^{\prime }(f(U_x))\subset \hat{N}\) , and let \(\hat{f}_x:=\bar{g}_x^{\prime }\circ f\circ \bar{g}_x^{-1}:\hat{N}\supset \hat{V}_x \rightarrow \hat{V}_x^{\prime }\subset \hat{N}\) , where \(\hat{V}_x:=\bar{g}_x(U_x)\) , \(\hat{V}_x^{\prime }:=\bar{g}_x^{\prime }(f(U_x))\) . Let \(f_x:=\alpha ^{-1}\circ \hat{f}_x\circ \alpha :N\supset V_x\rightarrow V_x^{\prime }\subset N\)  where \(V_x:=\alpha ^{-1}(\hat{V}_x)\) , \(V_x^{\prime }:=\alpha ^{-1}(\hat{V}_x^{\prime })\) .
By Corollary REF , for some \(\varepsilon _x\in \lbrace 0,1\rbrace \) , the map \(\tau ^{\varepsilon _x}\circ f_x\)  locally preserves the coset foliation of \(K_j\)  for all \(1\le j\le n-1\) . Now Lemma REF (4) gives that the map
\(\alpha \circ (\tau ^{\varepsilon _x}\circ f_x)\circ \alpha ^{-1}=(\alpha \circ \tau ^{\varepsilon _x}\circ \alpha ^{-1})\circ \hat{f}_x=\rho (\tau ^{\varepsilon _x})\circ \hat{f}_x\) 
locally preserves the fibration \(\pi _j\)  for \(1\le j\le n-1\) ; here we have used the fact that \(\rho (\tau )\circ \alpha =\alpha \circ \tau \) . Applying Proposition REF , we see that \(\rho (\tau ^{\varepsilon _x})\circ \hat{f}_x\)  agrees with some element of \(G\)  and therefore \(f=\rho (\Phi _x)\mbox{\Large $|$\normalsize }_{U_x}\)  for some \(\Phi _x\in \operatorname{Aut}(G)\) . Since \(\Phi _x\)  is locally constant as a function of \(x\) , by the connectedness of \(U\) , the automorphism \(\Phi _x\)  is independent of \(x\) , and so \(f=\rho (\Phi )\mbox{\Large $|$\normalsize }_U\)  for some \(\Phi \in \operatorname{Aut}(G)\) . By Lemma REF (1) we have \(\Phi =\Phi _0^{\varepsilon }\circ I_g\)  for some \(\varepsilon \in \lbrace 0,1\rbrace \) , \(g\in G\) , where \(\Phi _0\)  denotes transpose-inverse; then \(\rho (\Phi )=(\rho (\Phi _0))^{\varepsilon }\circ \rho (I_g)=\psi ^\varepsilon \circ g\)  using Lemma REF (5).


The complex and quaternionic cases
 
The arguments from the previous sections are also valid in the complex and quaternion cases, with some straightforward modifications. In this section we indicate what modifications are needed in these cases. The necessity for these modifications are due to the presence of nontrivial automorphisms of \(\mathbb {C}\)  and \(\mathbb {H}\)  and the non-commutativity of the quaternions.
We first recall some facts about quaternions. Given any quaternion
\(x=x_0+x_1i+x_2j+x_3k\in \mathbb {H}\)  (\(x_i\in \mathbb {R}\) ), the conjugation of \(x\)  is
\(\bar{x}=x_0-x_1i-x_2j-x_3k\) . It is easy to check that \(\overline{xy}=\bar{y}\bar{x}\)  for any \(x,y\in \mathbb {H}\) .
Let \(\lambda , \mu \)  be unit quaternions
satisfying \(\lambda ^2=\mu ^2=(\lambda \mu )^2=-1\) . Set \(\nu =\lambda \mu \) . Then we have
\(\mu =\nu \lambda \)  and \(\lambda =\mu \nu \) . Define a map
\(h=h_{\lambda , \mu , \nu }: \mathbb {H}\rightarrow \mathbb {H}\)  by
\(h(a_0+ia_1+ja_2+ka_3)=a_0+\lambda a_1+\mu a_2+\nu a_3.\) 
Then it is easy to check that \(h\)  is an automorphism of \(\mathbb {H}\) : it is a real linear isomorphism and \(h(xy)=h(x)h(y)\)  for any \(x, y\in \mathbb {H}\) . Conversely, for any automorphism \(h: \mathbb {H}\rightarrow \mathbb {H}\) , if we set
\(\lambda :=h(i)\) , \(\mu :=h(j)\) , \(\nu :=h(k)\) , then \(\lambda ^2=\mu ^2=\nu ^2=-1\) , \(\nu =\lambda \mu \)  and \(h=h_{\lambda , \mu , \nu }\) .
By the Skolem-Noether theorem, every automorphism of \(\mathbb {H}\)  is inner.

Changes needed for Section 2
Let \(F=\mathbb {C}, \mathbb {H}\) .
Let \(\operatorname{GL}(n,F)\)  be the group of invertible elements in the ring \(M_n(F)\)  of \(n\times n\)  matrices with entries in \(F\) .
The objects \(P_F^+, P_F^-\)  and \(N_F\)  are defined as before with \(\mathbb {R}\)  replaced by \(F\) . The group \(G_F\)  is defined as before in the complex case with \(\mathbb {R}\)  replaced with \(\mathbb {C}\) :
\(G_{\mathbb {C}}=\text{GL}(n, \mathbb {C})/{\lbrace \lambda \,\text{id}|0\ne \lambda \in \mathbb {C}\rbrace }.\) 
In the quarternion case it is defined by
\(G_{\mathbb {H}}=\text{GL}(n, \mathbb {H})/{\lbrace \lambda \,\text{id}\mid 0\ne \lambda \in \mathbb {R}\rbrace }.\) 
Note that
\(\lbrace aI\mid a\in \mathbb {H}\backslash \lbrace 0\rbrace \rbrace \)  is not normal in \(GL(n, \mathbb {H})\)  and so we cannot quotient out by this subgroup.
Similarly,
\(P_{\mathbb {C}}=P_{\mathbb {C}}^-/{\lbrace \lambda \,\text{id}\mid 0\ne \lambda \in \mathbb {C}\rbrace } \)  and
\(P_{\mathbb {H}}=P_{\mathbb {H}}^-/{\lbrace \lambda \,\text{id}\mid 0\ne \lambda \in \mathbb {R}\rbrace } . \)


The flag manifold
We view \(F^n\)  as a right \(F\)  module.
The flag manifold \({F}\)  is the set of (complete) flags in \(F^n\) , i.e. the collection of nested families of submodules of \(F^n\)
\(W_1\subset \ldots \subset W_{n-1}\) 
where \(W_j\)  has dimension (rank) \(j\) .
Matrix multiplication yields an action \(\operatorname{GL}(n,F)\operatorname{\curvearrowright }F^n\)  by \(F\) -module automorphisms in the usual way, which induces actions \(\operatorname{GL}(n,F)\operatorname{\curvearrowright }{k,F}\) ,
where \( {k,F}\)  is the Grassmannian of submodules of dimension (rank) \(k\) .
Lemma REF 
holds without changes.


Automorphisms
The map \(A\mapsto (A^*)^{-1}\)  is a Lie group automorphism of \(\operatorname{GL}(n,F)\) , where \(A^*\)  denotes the conjugate transpose of \(A\) .
So the map \(\tau : GL(n, F) \rightarrow GL(n, F)\)  given by \(\tau (A)=\Pi (A^*)^{-1}\Pi ^{-1}\)  is also an automorphism and
induces an automorphism (still denoted by \(\tau \) ) of \(\mathfrak {gl}(n, F)\)  which is given by
\((\tau (A))_{ij}=-\overline{A}_{n-j+1, n-i+1}.\) 
For any automorphism \(h\)  of \(F\) , the automorphism
\( GL(n, F) \rightarrow GL(n, F)\) , \((a_{ij})\mapsto (h(a_{ij}))\) 
of \(GL(n, F)\)  induces an automorphism of \(G\) , which we denote by
\(\hat{h}\) .
Theorem 6.1 ([7]}) 
Every automorphism of \(G_F\)  is induced by an automorphism of \(GL(n, F)\) .
The group
\(\text{Aut}(G_F)\)  is generated by \(\tau \) , maps of the form \(\hat{h}\)  (with
\(h\in Aut(F)\) ) and the inner automorphisms.

Notice that the automorphisms \(h\)  associated with
Lie group automorphisms are continuous.
We recall that there are only two continuous automorphism of \(\mathbb {C}\) : the identity map and the complex conjugation. On the other hand,
by the Skolem-Noether theorem, every automorphism \(h: \mathbb {H}\rightarrow \mathbb {H}\)  is inner.
It follows that the automorphism \(\hat{h}\)  of \(G_{\mathbb {H}}\) 
is also inner: if \(h=I_a\)  for some \(a\in \mathbb {H}\) , then \(\hat{h}=I_g\)  with
\(g=\text{diag}(a, \cdots , a)\) .
The following is the counterpart of
Lemma REF 
in the quaternion and complex cases.
Lemma 6.2 
Let \(F=\mathbb {C}, \mathbb {H}\) . Let \(n\ge 3\)  if \(F=\mathbb {H}\)  and \(n\ge 4\)  if \(F=\mathbb {C}\) .

If \(\Phi \in \operatorname{Aut}(G_F)\)  and \(\Phi (N_F)=N_F\) , then \(\Phi \)  induces a graded automorphism of \(N_F\)  if and only if \(\Phi \mbox{\Large $|$\normalsize }_{N_F}=\tau ^\varepsilon \circ \hat{h}\circ I_g\mbox{\Large $|$\normalsize }_{N_F}\)  for some \(\varepsilon \in \lbrace 0,1\rbrace \) , some continuous automorphism \(h\)  of \(F\)  and \(g=\operatorname{diag}(\lambda _1,\ldots ,\lambda _n)\in G_F\) .

Every graded automorphism of \(N_F\)  arises as in (1).

For \(1\le j\le n-1\)  let \(\mathfrak {k}_j\subset \mathfrak {n}_F\)  be the Lie subalgebra generated by \(\lbrace a X_{i,i+1}|a\in F, i\ne n-j\rbrace \) , and \(K_j\subset N_F\)  be the Lie subgroup with Lie algebra \(\mathfrak {k}_j\) . A graded automorphism \(N_F\rightarrow N_F\)  is induced by conjugation by some \(g=\operatorname{diag}(\lambda _1,\ldots ,\lambda _n)\)  if and only if it preserves the subgroups \(K_j\)  for \(1\le j\le n-1\) .


We remark that Lemma REF  (2) implies that every graded automorphism of \({\mathfrak {n}}_{n, \mathbb {C}}\)  is either complex linear or complex antilinear.
Let \(n\ge 3\) .
The proof of (1) and (3) are the same (by using Theorem REF )
as that of (1) and (3) in Lemma REF . Here we prove (2).
The proof is by induction on \(n\) . We first consider the case \(n=3\) .
Denote \(X=X_{12}\) , \(Y=X_{23}\)  and \(Z=X_{13}\) .
We have \([aX, bY]=abZ\)  for \(a, b\in \mathbb {H}\) . Note that the Lie bracket is not linear over \(\mathbb {H}\) .
Let \(\mathbb {H} X=\lbrace aX\mid a\in \mathbb {H}\rbrace \)  be the subspace spanned by \(X\) . It has dimension 4 over \(\mathbb {R}\) . Similarly we have \(\mathbb {H} Y\)  and \(\mathbb {H} Z\) .
The grading \(\mathfrak {n}_{3,\mathbb {H}}=V_1\oplus V_2\)  is given by
\(V_1=\mathbb {H} X\oplus \mathbb {H} Y\)  and \(V_2=\mathbb {H} Z\) .
Let \(A:\mathfrak {n}_{3,\mathbb {H}}\rightarrow \mathfrak {n}_{3,\mathbb {H}}\)  be a graded automorphism.
We claim that \(A\)  satisfies either \(A(\mathbb {H} X)= \mathbb {H} X\) ,
\(A(\mathbb {H} Y)= \mathbb {H} Y\)  or \(A(\mathbb {H} X)= \mathbb {H} Y\) ,
\(A(\mathbb {H} Y)= \mathbb {H} X\) . There are \(a, b\in \mathbb {H}\)  such that
\(A(X)=aX+bY\) . To prove the claim it suffices to show that \(a=0\)  or \(b=0\) . Suppose \(a, b\ne 0\)  we shall get a contradiction.
There are \(c, d\in \mathbb {H}\)  such that \(A(iX)=cX+dY\) .
As \([X, iX]=0\)  we have \(0=[A(X), A(iX)]=(ad-cb)Z\)  (being careful about the order in \(cb\) ), which yields \(a^{-1}c=db^{-1}\) .
Set \(\lambda =a^{-1}c\) . We have \(c=a\lambda \)  and \(d=\lambda b\)  and so
\(A(iX)=a\lambda X+\lambda b Y\) . Similarly there are \(\mu , \nu \in \mathbb {H}\)  such that
\(A(jX)=a\mu X+\mu b Y\)  and \(A(kX)=a\nu X+\nu b Y\) . By further considering
the brackets between \(A(iX)\) , \(A(jX)\) , \(A(kX)\)  we get
\(a\lambda \mu b=a\mu \lambda b\) , \(a\lambda \nu b=a\nu \lambda b\)  and
\(a \mu \nu b=a\nu \mu b\) . It follows that \(\lambda \) , \(\mu \)  and \(\nu \)  commute with each other. Recall the fact that two quaternions commute with each other if and only if their imaginary parts are real multiples of each other. Hence there are real numbers
\(r_i, t_i\)  (\(i=1, 2,3\) ) and a purely imaginary quaternion \(h\)  such that
\(\lambda =r_1+t_1 h\) , \(\mu =r_2+t_2 h\) , \(\nu =r_3+t_3 h\) . We then have
\(A(iX)=r_1(aX+bY)+t_1(ahX+hbY)\)  and so \(A(iX)\)  lies in the 2-dimensional real vector subspace spanned by \(aX+bY\)  and \(ahX+hbY\) . Similarly \(A(jX)\)  and \(A(kX)\)  also lie in this subspace, contradicting the fact that \(A\)  is an isomorphism. This finishes the proof of the claim.
After possibly composing \(A\)  with \(\tau \)  we may assume that \(A(\mathbb {H} X)=\mathbb {H} X\)  and
\(A(\mathbb {H} Y)=\mathbb {H} Y\) .
There are \(a_0, a_1, a_2, a_3, b_0, b_1, b_2, b_3\in \mathbb {H}\)  such that
\(A(X)=a_0X\) , \(A(iX)=a_1X\) , \(A(jX)=a_2X\) , \(A(kX)=a_3X\)  and
\(A(Y)=b_0Y\) , \(A(iY)=b_1Y\) , \(A(jY)=b_2Y\) , \(A(kY)=b_3Y\) . By applying \(A\)  to
\([X, iY]=[iX, Y]=[jX, kY]=-[kX, jY]\) ,
\([X, jY]=[jX, Y]=[kX, iY]=-[iX, kY]\)  and \([X, kY]=[kX, Y]=[iX, jY]=-[jX, iY]\)  we obtain
\(a_0b_1=a_1b_0=a_2b_3=-a_3b_2,\) 

\(a_0b_2=a_2b_0=a_3b_1=-a_1b_3,\) 

\(a_0b_3=a_3b_0=a_1b_2=-a_2b_1.\) 
From these we get
\(a_0^{-1}a_1=b_1b_0^{-1}=-b_2b_3^{-1}=b_3b_2^{-1},\) 

\(a_0^{-1}a_2=b_1b_3^{-1}=b_2b_0^{-1}=-b_3b_1^{-1},\) 

\(a_0^{-1}a_3=-b_1b_2^{-1}=b_2b_1^{-1}=b_3b_0^{-1}.\) 
Set \(\lambda =a_0^{-1}a_1\) , \(\mu =a_0^{-1}a_2\)  and \(\nu =a_0^{-1}a_3\) .
From \(\lambda =-b_2b_3^{-1}=b_3b_2^{-1},\)  we get \(\lambda ^2=-1\) . Similarly from \(\mu =b_1b_3^{-1}=-b_3b_1^{-1},\)  we get \(\mu ^2=-1\) . Finally
\(\nu =b_3b_0^{-1}=b_3b_2^{-1} b_2b_0^{-1}=\lambda \mu \) .
Also notice \(a_1=a_0\lambda \) , \(a_2=a_0\mu \) , \(a_3=a_0\nu \)  and
\(b_1=\lambda b_0\) , \(b_2=\mu b_0\) , \(b_3=\nu b_0\) . So the automorphism \(A\)  is given by \(A(X)=a_0X\) , \(A(iX)=a_0\lambda X\) , \(A(jX)=a_0\mu X\) , \(A(kX)=a_0\nu X\)  and
\(A(Y)=b_0Y\) , \(A(iY)=\lambda b_0Y\) , \(A(jY)=\mu b_0 Y\) , \(A(kY)=\nu b_0 Y\) .
Now it is easy to check that \(A=\text{Ad}_g\mbox{\Large $|$\normalsize }_{\mathfrak {n}_\mathbb {H}}\circ \hat{h}_{\lambda , \mu ,\nu }\) , where \(g=\text{diag}(a_0, 1, b_0^{-1})\) .
Now assume \(n\ge 4\) . By an argument similar to the real case (using rank and the induction hypothesis), we get (after possibly composing with \(\tau \) ) \(A(\mathbb {H} X_{i, i+1})=\mathbb {H} X_{i, i+1}\)  for each \(1\le i\le n-1\) . Then there are nonzero quaternions \(a_1, \cdots , a_{n-1}\)  such that
\(A(X_{i, i+1})=a_iX_{i, i+1}\) . Set \(b_n=1\)  and \(b_i=(a_i\cdots a_{n-1})^{-1}\)  for \(1\le i\le n-1\) , and let
\(g=\text{diag}(b_1, \cdots , b_n)\) .
By composing \(A\)  with \(\text{Ad}_g\mbox{\Large $|$\normalsize }_{\mathfrak {n}_\mathbb {H}}\)  we may assume that
\(A(X_{i,i+1})=X_{i, i+1}\)  for each \(1\le i\le n-1\) .
Now for each \(1\le i\le n-2\) , \(\mathbb {H} X_{i, i+1}\)  and \(\mathbb {H} X_{i+1, i+2}\)  generate a Lie subalgebra isomorphic to \(\mathfrak {n}_3\) . By the previous paragraph we see that there are \(\lambda _i, \mu _i\)  satisfying \(\lambda _i^2=\mu _i^2=(\lambda _i\mu _i)^2=-1\)  such that
\(A((a_0+a_1i+a_2j+a_3k)X)=(a_0+a_1\lambda _i+a_2\mu _i+a_3\nu _i)X\)  for \(X=X_{i,i+1}, X_{i+1, i+2}, X_{i, i+2}\) , where \(\nu _i=\lambda _i\mu _i\)  and \(a_0, a_1, a_2, a_3\in \mathbb {R}\) . By considering the Lie subalgebra generated by \(X_{i+1, i+2}\)  and
\(X_{i+2, i+3}\)  and comparing the values for
\(A((a_0+a_1i+a_2j+a_3k)X_{i+1, i+2})\) 
we get \(\lambda _i=\lambda _{i+1}\) , \(\mu _i=\mu _{i+1}\)  and \(\nu _i=\nu _{i+1}\) . It follows that
\(A=\hat{h}_{\lambda , \mu , \nu }\)  with \(\lambda =\lambda _1\) , \(\mu =\mu _1\)  and \(\nu =\lambda \mu \) .

Let \(n\ge 4\) .
The proofs of (1) and (3) are the same as in the real case by using Theoem REF . As remarked above, the automorphism \(\hat{h}\)  is either the identity map or the complex conjugation.
(2) The proof is by induction on \(n\) . We first consider the case \(n=4\) .
Let \(A: {\mathfrak {n}}_{4, \mathbb {C}} \rightarrow {\mathfrak {n}}_{4, \mathbb {C}}\)  be a
graded automorphism. We observe that \({\mathfrak {n}}_{n, \mathbb {C}}\)  is the complexification of \(\mathfrak {n}_n\) . An easy calculation shows
\(\text{rank}(\text{ad}\, x)=\dim (\text{ad}\, x( {\mathfrak {n}}_{4, \mathbb {C}}))\ge 4\)  for any nonzero element \(x\)  in the first layer of \( {\mathfrak {n}}_{4, \mathbb {C}} \) . Clearly,
\(\text{rank}(\text{ad}\, x)\le 6\) . So the condition
\(\max \lbrace \text{rank}(\text{ad}\, x)|x\in V_1\rbrace <2 \min \lbrace \text{rank}(\text{ad}\, x)|0\ne x\in V_1\rbrace \) 
in [2]} is satisfied and we conclude that
every graded automorphism of \({\mathfrak {n}}_{4, \mathbb {C}}\)  is either complex linear or complex antilinear. So after possibly composing \(A\)  with the complex conjugation we may assume \(A\)  is complex linear. The rest of the arguemnt in the case \(n=4\)  is the same as in the real case.
Now let \(n\ge 5\)  and assume the statement holds for all integers less than \(n\) .
Let \(A: {\mathfrak {n}}_{n, \mathbb {C}} \rightarrow {\mathfrak {n}}_{n, \mathbb {C}}\)  be a
graded automorphism.
Arguing as in the real case using rank, we see that after possibly composing
\(A\)  with \(\tau \)  we have \(A(\mathbb {C} X_{i, i+1})=\mathbb {C} X_{i, i+1}\)  for each \(i\) .
So \(A (\mathfrak {n}_+)=\mathfrak {n}_+\)  and \(A (\mathfrak {n}_-)=\mathfrak {n}_-\) ,
where \(\mathfrak {n}_+\)  is the Lie sub-algebra of \( {\mathfrak {n}}_{n, \mathbb {C}}\)  generated by \(\lbrace X_{i, i+1}, 1\le i\le n-2\rbrace \)  and
\(\mathfrak {n}_-\)  is the Lie sub-algebra of \( {\mathfrak {n}}_{n, \mathbb {C}}\)  generated by \(\lbrace X_{i, i+1}, 2\le i\le n-1\rbrace \) . Since
\(\mathfrak {n}_+\)  and \(\mathfrak {n}_-\)  are isomorphic to
\({\mathfrak {n}}_{n-1, \mathbb {C}} \) , the induction hypothesis implies that each of
\(A|_{\mathfrak {n}_+}\) , \(A|_{\mathfrak {n}_+}\)  is either complex linear or complex antilinear. Since \(\mathfrak {n}_+\)  and \(\mathfrak {n}_-\)  have nontrivial intersection,
either both \(A|_{\mathfrak {n}_+}\) , \(A|_{\mathfrak {n}_+}\)  are complex linear or both are complex antilinear. Hence after possibly composing \(A\)  with the complex conjugation we may assume \(A\)  is complex linear. As \(A\)  also satisfies
\(A(\mathbb {C} X_{i, i+1})=\mathbb {C} X_{i, i+1}\)  for all \(i\) , we see that \(A=\text{Ad}_g\)  for some \(g=\text{diag}(\lambda _1, \cdots , \lambda _n)\in G_{\mathbb {C}}\) .
This finishes the proof of (2).
The “Hermitian product” on \(\mathbb {H}^n\)  is defined by
\(<z,w>=\sum _i\bar{z_i}w_i\)  for \(z=(z_i), w=(w_i)\in \mathbb {H}^n\) .
Then one can check by direct calculation that \(\overline{<z,w>}=<w,z>\)  and \(<A^*z, w>=<z, Aw>\)  for \(z,w\in \mathbb {H}^n\)  and \(A\in M_n(\mathbb {H})\) .
For any \(\mathbb {H}\) -linear subspace (submodule) \(W\)  of \(\mathbb {H}^n\) , the “orthogonal complement” \(W^\perp \)  is defined by \(W^\perp =\lbrace z\in \mathbb {H}^n| <z,w>=0\; \forall w\in W\rbrace \) .
Lemma REF  holds for \(F=\mathbb {C}, \mathbb {H}\)  where in (5) the automorphism
\(\Phi _0\)  is induced by the map \(A\mapsto (A^*)^{-1}\) .
The proof of Lemma REF  (5) goes through in the quaternion case since
\(<(g^{-1})^*W^+_j, gW^-_{n-j}>=<W^+_j, g^{-1}gW^-_{n-j}>=0.\) 
Of course, the proof is also valid in the complex case if we use the standard Hermitian product in \(\mathbb {C}^n\) .
Lemmas REF  and REF  hold in the complex and quaternion cases without change.


Changes needed for Section 3
Lemma REF  and
Corollary REF 
hold in the complex case for \(n\ge 4\)  and in the quaternion case for \(n\ge 3\) . Note that the analog of Lemma REF  in the \(F= case fails when \) n=3\(, as in the real \) n=3\( case. We indicate the changes needed in the quaternion case below. We skip the complex case since it is similar. Alternatively the complex case also follows from Corollary 8.2 of \cite {KMX1} as by Lemma\ref {graded-quaternion} (2)every graded automorphism of \) nn, C\( is either complex linear or complex antilinear.\)
Lemma 6.3 
Let \(n\ge 3\) .
Let \(U\subset N_{\mathbb {H}}\)  be a connected open subset, and for some \(p>\nu \)  let \(f:N_{\mathbb {H}}\supset U\rightarrow N_{\mathbb {H}}\)  be a \(W^{1,p}_{\operatorname{loc}}\) -mapping whose Pansu differential is an isomorphism almost everywhere. Then after possibly composing with \(\tau \) , if necessary,
for a.e. \(x\in U\)  the Pansu differential \(D_Pf(x)\)  preserves the subspace \(\mathbb {H}X_{i,i+1}\subset V_1\)  for every \(1\le i\le n-1\) .

Here we indicate the differential forms used in the calculations, the rest of the argument being the same.
For \(1\le s<t\le n\) , denote \(Y_{st}=iX_{st}\) , \(Z_{st}=jX_{st}\) ,
\(W_{st}=kX_{st}\) . Then \(\lbrace X_{st}, Y_{st}, Z_{st}, W_{st} |1\le s< t\le n\rbrace \)  form a basis of left invariant vector fields on \(N\) . The only nontrivial bracket relations between the basis elements are given by (for \(1\le s_1<s_2<s_3\le n\) ):
\(-[X_{s_1 s_2}, X_{s_2 s_3}]=[Y_{s_1 s_2}, Y_{s_2 s_3}]=[Z_{s_1 s_2}, Z_{s_2 s_3}]=[W_{s_1 s_2}, W_{s_2 s_3}]=-X_{s_1s_3}\) 

\([X_{s_1 s_2}, Y_{s_2 s_3}]=[Y_{s_1 s_2}, X_{s_2 s_3}]=[Z_{s_1 s_2}, W_{s_2 s_3}]=-[W_{s_1 s_2}, Z_{s_2 s_3}]=Y_{s_1s_3}\) 

\([X_{s_1 s_2}, Z_{s_2 s_3}]=[Z_{s_1 s_2}, X_{s_2 s_3}]=[W_{s_1 s_2}, Y_{s_2 s_3}]=-[Y_{s_1 s_2}, W_{s_2 s_3}]=Z_{s_1s_3}\) 

\([X_{s_1 s_2}, W_{s_2 s_3}]=[W_{s_1 s_2}, X_{s_2 s_3}]=[Y_{s_1 s_2}, Z_{s_2 s_3}]=-[Z_{s_1 s_2}, Y_{s_2 s_3}]=W_{s_1s_3}. \) 
Let \(\alpha _{st}\) , \(\beta _{st}\) , \(\gamma _{st}\) , \(\eta _{st}\)  be the dual basis of left invariant 1-forms. We have
\(d\alpha _{s_1s_3}=\sum _{s_1<s_2<s_3} (-\alpha _{s_1s_2}\wedge \alpha _{s_2s_3}+\beta _{s_1s_2}\wedge \beta _{s_2s_3}+\gamma _{s_1s_2}\wedge \gamma _{s_2s_3}+\eta _{s_1s_2}\wedge \eta _{s_2s_3} ) \) 

\(d\beta _{s_1s_3}=\sum _{s_1<s_2<s_3} (-\alpha _{s_1s_2}\wedge \beta _{s_2s_3}-\beta _{s_1s_2}\wedge \alpha _{s_2s_3}-\gamma _{s_1s_2}\wedge \eta _{s_2s_3}+\eta _{s_1s_2}\wedge \gamma _{s_2s_3} ) \) 

\(d\gamma _{s_1s_3}=\sum _{s_1<s_2<s_3} (-\alpha _{s_1s_2}\wedge \gamma _{s_2s_3}-\gamma _{s_1s_2}\wedge \alpha _{s_2s_3}-\eta _{s_1s_2}\wedge \beta _{s_2s_3}+\beta _{s_1s_2}\wedge \eta _{s_2s_3} ) \) 

\(d\eta _{s_1s_3}=\sum _{s_1<s_2<s_3} (-\alpha _{s_1s_2}\wedge \eta _{s_2s_3}-\eta _{s_1s_2}\wedge \alpha _{s_2s_3}-\beta _{s_1s_2}\wedge \gamma _{s_2s_3}+\gamma _{s_1s_2}\wedge \beta _{s_2s_3} ). \) 
We pull back the following closed left invariant forms
\(\omega _+=\bigwedge _{2\le s\le n} (\alpha _{1s}\wedge \beta _{1s}\wedge \gamma _{1s}\wedge \eta _{1s})\) 

\(\omega _-=\bigwedge _{1\le s\le n-1} (\alpha _{sn}\wedge \beta _{sn}\wedge \gamma _{sn}\wedge \eta _{sn}).\) 
By Lemma REF  on graded automorphisms, the pull-back has the form \(f_P^*\omega _+=u_+\omega _++u_-\omega _-\)  as before.
Let
\(\eta _-=\bigwedge _{2\le s<t\le n} (\alpha _{st}\wedge \beta _{st}\wedge \gamma _{st}\wedge \eta _{st})\) 

\(\eta _+=\bigwedge _{1\le s<t\le n-1} (\alpha _{st}\wedge \beta _{st}\wedge \gamma _{st}\wedge \eta _{st}).\) 
We apply the pull-back theorem to
\(f_P^*\omega _+\)  and \(\eta =i_X\eta _-\) , \(i_X\eta _+\) , where
\(X\in \lbrace X_{s (s+1)}, Y_{s(s+1)}, Z_{s(s+1)}, W_{s(s+1)}\rbrace \)  and \(i_X\)  denotes the interior product with respect to \(X\) .
As before, this yields \(Xu_+=0\)  for
\(X\in \lbrace X_{s (s+1)}, Y_{s(s+1)}, Z_{s(s+1)}, W_{s(s+1)}\rbrace \)  with \(2\le s\le n-1\)  and \(Xu_-=0\)  for
\(X\in \lbrace X_{s (s+1)}, Y_{s(s+1)}, Z_{s(s+1)}, W_{s(s+1)}\rbrace \)  with \(1\le s\le n-2\) . The rest of the argument is the same as in the real case.


Changes needed for Section 4
In the quaternion case, we note that the action of \(PGL(n, \mathbb {H})\)  on the projective frames is still transitive, but is no longer free. The reason is that \(a I_n\)  defines a nontrivial element in
\(PGL(n, \mathbb {H})\)  for \(a\in \mathbb {H}\setminus \mathbb {R}\) , but fixes the standard projective frame.
Below is a version of Lemma REF  for the quaternion case.
A similar statement holds for the complex case. A line in \(\mathbb {H}^2\)  is a subset of the form \(\lbrace (a_1, a_2)x+(b_1, b_2)|x\in \mathbb {H}\rbrace \)  for some \((b_1, b_2)\in \mathbb {H}^2\) ,
\((0,0)\ne (a_1, a_2)\in \mathbb {H}^2\) . This line is said to be parallel to \((a_1, a_2)=a_1e_1+a_2e_2\) .
Lemma 6.4 
Suppose \(U\subset \mathbb {H}^2\)  is a connected open subset, and
\(\phi =(\phi _1,\phi _2):\mathbb {H}^2\supset U\rightarrow \mathbb {H}^2\) 
is a continuous map such that for every \(v\in \lbrace e_1,e_2,e_1+e_2\rbrace \)  and every line \(L\)  parallel to \(v\) , the image of \(L\cap U\)  is contained in a line parallel to \(v\) . Assume further that for each \(q\in \lbrace i, j, k\rbrace \) , lines parallel to
\(e_1+qe_2\)  are mapped into lines (not necessarily parallel to \(e_1+qe_2\) ).
Then \(\phi \)  is of the form
\(\phi (x,y)=(a h(x)+b_1, a h(y)+b_2)\) 
where \(a, b_1,b_2\in \mathbb {H}\)  and \(h: \mathbb {H}\rightarrow \mathbb {H}\)  is either an automorphism of \(\mathbb {H}\)  or the zero map.

The argument of Lemma REF  yields that
\(\phi \)  is of the form
\(\phi (x,y)=(m(x)+b_1, m(y)+b_2)\) , where
\(b_1,b_2\in \mathbb {H}\)  and \(m:\mathbb {H}\rightarrow \mathbb {H}\)  is a real linear map.
We shall show that either \(m\)  is the zero map or there is some automorphism \(h\)  of \(\mathbb {H}\)  and some
\(a\in \mathbb {H}\)  such that \(m(x)=a h(x)\) . We may assume \(b_1=b_2=0\)  after possibly composing \(\phi \)  with a translation. Then the assumption implies that the line \((1,i)\mathbb {H}\)  is mapped
by \(\phi \)  into a line \((a_1, a_2)\mathbb {H}\)  (at least one of \(a_1, a_2\)  is nonzero) through the origin. There are \(t_1, t_2\in \mathbb {H}\)  such that
\((m(1), m(i))=\phi (1,i)=(a_1t_1, a_2t_1)\)  and \((m(i), m(-1))=\phi (i, -1)=(a_1t_2, a_2t_2)\) . By comparing the components we get \(m(i)=a_2t_1=a_1t_2\) , \(m(1)=a_1t_1=-a_2t_2\) . Suppose \(m(1)=0\) . As at least one of \(a_1, a_2\)  is nonzero we have \(t_1=0\)  or \(t_2=0\) , which implies \(m(i)=0\) . Similarly \(m(j)=m(k)=0\) . In this case \(m\)  is the zero map.
Now we assume \(m(1)\ne 0\) .
Then there is some \(\lambda \in \mathbb {H}\)  such that
\(\phi (i, -1)=\phi (1,i)\lambda \) . By comparing the two components of both sides we get
\(m(i)=m(1)\lambda \)  and \(-m(1)=m(-1)=m(i)\lambda \) , which yields
\(\lambda ^2=-1\) . Similarly by considering the lines \((1,j)\mathbb {H}\)  and
\((1,k)\mathbb {H}\)  we see that there are \(\mu \)  and \(\nu \)  satisfying \(\mu ^2=\nu ^2=-1\)  such that \(m(j)=m(1)\mu \) , \(-m(1)=m(-1)=m(j)\mu \) ,
\(m(k)=m(1)\nu \)  and \(-m(1)=m(-1)=m(k)\nu \) .
Since \((j,k)=(1,i)j\in (1,i)\mathbb {H}\) , there is some \(c\in \mathbb {H}\)  such that
\(\phi (j,k)=\phi (1,i) c\) . This gives us \(m(j)= m(1) c\)  and \(m(k)=m(i)c\) . As we also have \(m(i)=m(1)\lambda \) ,
\(m(j)=m(1)\mu \)  and \(m(k)=m(1)\nu \) , we conclude \(c=\mu \)  and \(\nu =\lambda \mu \) . The three numbers \(\lambda , \mu ,\nu \)  satisfy \(\lambda ^2=\mu ^2=\nu ^2=-1\)  and \(\nu =\lambda \mu \) .
As \(m\)  is \(\mathbb {R}\) -linear, for any \(x=x_0+ix_1+jx_2+kx_3\)  (\(x_i\in \mathbb {R}\) ) we get \(m(x)=m(1) h_{\lambda , \mu , \nu }(x)\) .
The arguments in Section  show that we may assume the map \(\phi \)  sends
lines parallel to \(v\in \lbrace e_1, e_2, e_1+e_2\rbrace \) 
into lines parallel to \(v\) . We claim that for any \(q\in \lbrace i,j,k\rbrace \) , lines parallel to \(e_1+qe_2\)  are mapped by \(\phi \)  into a family of parallel lines (not necessarily parallel to \(e_1+qe_2\) ). To see this, we notice that for a suitable diagonal matrix \(g=\text{diag}(a_1, a_2, 1)\in GL(3, \mathbb {H})\) , \(g\circ \hat{f}_1\)  satisfy \(g\circ \hat{f}_1(\text{span}(e_1+qe_2+e_3))=\text{span}(e_1+qe_2+e_3)\) 
and \(g\circ \hat{f}_1(\hat{W}^i_1)=\hat{W}^i_1\)  for \(i=1,2,3\) .
Since \(\mathbb {H}^3\)  is a right \(\mathbb {H}\)  module, here
for any \((x_1, x_2, x_3)\in \mathbb {H}^3\) , \(\operatorname{span}(x_1,x_2, x_3)=\lbrace (x_1x, x_2x, x_3x)|x\in \mathbb {H}\rbrace \) .
Then it follows that
\(\bar{g}\circ \phi \)  sends lines parallel to \(e_1+qe_2\)  to lines parallel to
\(e_1+qe_2\) , where \(\bar{g}:\mathbb {H}^2\rightarrow \mathbb {H}^2\)  is the linear map given by the diagonal matrix \(\text{diag}(a_1, a_2)\) .
Consequently \(\phi \)  sends lines parallel to \(e_1+qe_2\)  into a family of parallel lines.

.
Only the last paragraph and the third last paragraph of the proof of Lemma REF 
need some changes.
As before we have \(\hat{f}_1([e_3])=[e_3]\) . Hence
for small \(r>0\)  we may define \(\phi :\mathbb {H}^2\supset B(0,r)\rightarrow \mathbb {H}^2\)  by \(\operatorname{span}(\phi (x_1,x_2),1)=\hat{f}_1(\operatorname{span}(x_1,x_2,1))\) . The fact
\(\hat{f}_1([e_3])=[e_3]\)  implies \(\phi (0,0)=(0,0)\) .
By Lemma REF 
the map \(\phi : \mathbb {H}^2\supset B(0,r) \rightarrow \mathbb {H}^2\)  in this case has the form
\(\phi (x_1, x_2)=(a h(x_1)+b_1, a h(x_2)+b_2)\) 
where \(a, b_1,b_2\in \mathbb {H}\)  and \(h: \mathbb {H}\rightarrow \mathbb {H}\)  is either an automorphism of \(\mathbb {H}\)  or the zero map.
As \(\phi (0,0)=(0,0)\) , we have \(b_1=b_2=0\)  and so
\(\phi (x_1, x_2)=(a h(x_1), a h(x_2))\) .
The argument for \(m\ne 0\)  applies here and shows that \(h\)  is an automorphism. Since any automorphism of \(H\)  is inner,
\(h(x)=bxb^{-1}\)  for some \(0\ne  b\in \mathbb {H}\)  and so
\(\phi (x_1, x_2)=(a bx_1b^{-1}, a bx_2b^{-1})\) .
Let \(\hat{g}:=\operatorname{diag}(ab, ab, b)\in \operatorname{PGL}(3,\mathbb {H})\) . Then \(\hat{g}_1:1\rightarrow 1\)  agrees with \(\hat{f}_1\)  near \([e_3]\) . Since \(\hat{g}\)  and \(\hat{f}\)  are both fibration-preserving, they agree near \(x=(W_1^-,W_2^-)\) .


Changes needed for Section 5
The counterpart of Theorem REF  for the complex and quaternion cases is:
Theorem 6.6 
Let \(U\subset F\)  be a connected open subset, and \(f:U\rightarrow F\)  be a \(W^{1,p}_{\operatorname{loc}}\) -mapping for \(p>\nu \) , such that the Pansu differential is an isomorphism almost everywhere.

If \(F=\mathbb {C}\)  and \(n\ge 4\) , then \(f\)  is the restriction of a diffeomorphism \( of the form \) 1C2 g\( where \) gPGL(n,\(, \) i{0,1}\( and \) C\( is complex conjugation. \newline \item If \) F=H\( and \) n3\(, then \) f\( is the restriction of a diffeomorphism \)  of the form \(\psi ^{\varepsilon } \circ g\)  where \(g\in \operatorname{PGL}(n,\mathbb {H})\) , \(\varepsilon \in \lbrace 0,1\rbrace \) .


The proof of Theorem REF  is the same as that of Theorem REF  except in the complex case where we may need to compose \(f\)  with the complex conjugation if necessary.


Global quasiconformal homeomorphisms
In this section we identify all global nondegenerate Sobolev maps \(N\rightarrow N\) . These are exactly the graded affine maps of \(N\) . This result is an immediate consequence of
Theorem REF .
An affine map of a Lie group \(G\)  is a map of the form \(L_g\circ \phi \) , where \(\phi \)  is an automorphism of \(G\)  and \(L_g\)  is left translation by \(g\in G\) . A graded affine map of a Carnot group \(N\)  is an affine map where the automorphism is a graded automorphism of \(N\) .
The following result applies to global quasiconformal homeomorphisms since quasiconformal maps are nondegenerate Sobolev maps.
Theorem 7.1  
Let \(N\)  be the Iwasawa group of \(\text{GL}(n, F)\) , with \(n\ge 4\)  for \(F=\mathbb {R}, \mathbb {C}\)  and \(n\ge 3\)  for \(F=\mathbb {H}\) .
Suppose \(f:N\rightarrow N\)  is a \(W^{1,p}_{\operatorname{loc}}\) -mapping for \(p>\nu \) , such that the Pansu differential is an isomorphism almost everywhere. Then \(f\)  is a graded affine map of \(N\) .

Let \(f\)  be as above. We shall show that there is a graded affine map \(f_0\)  such that
\(f_0^{-1}\circ f\)  is the identity map.
After replacing \(f\)  with \(L_{f(0)^{-1}}\circ f\)  we may assume \(f(0)=0\) .
We identify \(N\)  with \(\hat{N}\)  and view \(f\)  as a map \(\mathcal {F}\supset \hat{N}\rightarrow \hat{N}\subset \mathcal {F}\) .
We first consider the case \(F=\mathbb {R}\) .
By Theorem REF , \(f\)  is the restriction to \(\hat{N}\)  of a map of the form
\(\psi ^{\varepsilon }\circ g\)  where \(g\in \operatorname{GL}(n,\mathbb {R})\) , \(\varepsilon \in \lbrace 0,1\rbrace \) .
Recall that the automorphism \(\tau =I_{\Pi }\circ \Phi _0\)  of \(\text{GL}(n, \mathbb {R})\) 
induces a graded automorphism (again denoted by \(\tau \) ) of \(N=\hat{N}\)  and acts on \(\mathcal {F}\)  as \(\Pi \circ \psi \) . By replacing \(f\)  with \(\tau \circ f\)  if necessary (when \(\epsilon =1\) ),
we may assume \(\varepsilon =0\)  and so \(f=g|_{\hat{N}}\) . This implies \(g\in P^-\)  as the stabilizer of 0 is \(P^-\) . So \(g=(g_{ij})\) 
is a lower triangular matrix. Now we can further assume that the entries on the diagonal of \(g\)  are 1, after replacing \(g\)  with \(D^{-1}g\) , where \(D=\text{diag}(g_{11}, \cdots , g_{nn})\) .
So now \(g\)  is a lower triangular matrix with 1s on the diagonal and such that
\(g(\hat{N})=\hat{N}\) . We next show that \(g=I_n\) .
Suppose \(g\ne I_n\) . We shall find a flag \(F\in \hat{N}\)  such that \(g(F)\notin \hat{N}\) ,
contradicting \(g(\hat{N})=\hat{N}\) . Let \(1\le k\le n-1\)  be the integer such that
\(g_{ij}=0\)  for all \(i>j>k\)  and \(g_{jk}\ne 0\)  for some \(j>k\) . Let \(j_0>k\)  be such that \(g_{j_0k}\ne 0\)  and \(g_{jk}=0\)  for all \(j>j_0\) . Denote
\(v_{j_0}=e_k-\sum _{j=k+1}^{j_0} g_{jk} e_j\) .
Let \(F=\lbrace W_j\rbrace \)  be the flag defined by \(W_j=W^-_j\)  for \(1\le j\le n-j_0\)  and \(n-k<j\le n\) , \(W_{n-j_0+1}=\text{span}\lbrace e_n, \cdots , e_{j_0+1}, v_{j_0}\rbrace \)  and \(W_j=\text{span}\lbrace e_n, \cdots , e_{j_0+1}, v_{j_0}, e_{j_0-1}, \cdots , e_{n-j+1}\rbrace \)  for \(n-j_0+2\le j\le n-k\) .
It is straightforward to check that \(F\in \hat{N}\) . However,
\(g(v_{j_0})=e_k\) 
and
\(g(W_{n-j_0+1})\cap W^+_{j_0-1}\)  contains the nontrivial element \(e_k\)  and so \(g(F)\notin \hat{N}\) .
The proof in the complex and quaternion cases are the same except we use Theorem REF  and in the complex case we may need to compose \(f\)  with the complex conjugation.
